Relative Minimums and Maximums - AP Calculus BC
Card 0 of 65
Find and classify all the critical points for
.
Find and classify all the critical points for .
First thing we need to do is take partial derivatives.





Now we want to find critical points, we do this by setting the partial derivative in respect to x equal to zero.

Now we want to plug in these values into the partial derivative in respect to y and set it equal to zero.




Lets summarize the critical points:
If 

If 

Now we need to classify these points, we do this by creating a general formula
.
, where
, is a critical point.
If
and
, then there is a relative minimum at 
If
and
, then there is a relative maximum at 
If
, there is a saddle point at 
If
then the point
may be a relative minimum, relative maximum or a saddle point.

Now we plug in the critical values into
.



Since
and
,
is a relative minimum.


Since
,
is a saddle point.


Since
,
is a saddle point


Since
,
is a saddle point
First thing we need to do is take partial derivatives.
Now we want to find critical points, we do this by setting the partial derivative in respect to x equal to zero.
Now we want to plug in these values into the partial derivative in respect to y and set it equal to zero.
Lets summarize the critical points:
If
If
Now we need to classify these points, we do this by creating a general formula .
, where
, is a critical point.
If and
, then there is a relative minimum at
If and
, then there is a relative maximum at
If , there is a saddle point at
If then the point
may be a relative minimum, relative maximum or a saddle point.
Now we plug in the critical values into .
Since and
,
is a relative minimum.
Since ,
is a saddle point.
Since ,
is a saddle point
Since ,
is a saddle point
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 







There is only one critical point and it is at
. We need to determine if this critical point is a maximum or minimum using
and
.



Since
and
,
is a relative minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There is only one critical point and it is at . We need to determine if this critical point is a maximum or minimum using
and
.
Since and
,
is a relative minimum.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




There are two possible values of
,
and
.
We find the corresponding values of
using
(found by rearranging the first derivative)

![y=\frac{3\left ( \frac{\sqrt[3]{2}}{3} \right )^2}{2} =\frac{3\left ( \frac{\sqrt[3]{4}}{9} \right )}{2} =\frac{\sqrt[3]{4}}{6}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784127/gif.latex)
There are critical points at
and
. We need to determine if the critical points are maximums or minimums using
and
.


At
,

Since
,
is a saddle point.
At
,
![D(x,y)=144\left ( \frac{\sqrt[3]{2}}{3} \right )\left ( \frac{\sqrt[3]{4}}{6} \right )-4=144\left ( \frac{\sqrt[3]{8}}{18} \right )-4](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784132/gif.latex)

![f_{xx}(x,y)=6\left ( \frac{\sqrt[3]{2}}{3} \right )=2\sqrt[3]{2}>0](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784134/gif.latex)
Since
and
,
is a relative minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There are two possible values of ,
and
.
We find the corresponding values of using
(found by rearranging the first derivative)
There are critical points at and
. We need to determine if the critical points are maximums or minimums using
and
.
At ,
Since ,
is a saddle point.
At ,
Since and
,
is a relative minimum.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




Squaring both sides of the equation gives us

Multiplying both sides of the equation by
gives us

There are three possible values of
;
,
and
.
We find the corresponding values of
using
(found by rearranging the first derivative)




There are critical points at
,
and
. We need to determine if the critical points are maximums or minimums using
and
.



At
,

Since
,
is a saddle point.
At
,


Since
,
is a saddle point.
At
,



Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Squaring both sides of the equation gives us
Multiplying both sides of the equation by gives us
There are three possible values of ;
,
and
.
We find the corresponding values of using
(found by rearranging the first derivative)
There are critical points at ,
and
. We need to determine if the critical points are maximums or minimums using
and
.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 











There is only one real value of
; 
We find the corresponding value of
using
(found by rearranging the first derivative)

There is a critical point at
. We need to determine if the critical point is a maximum or minimum using
and
.



At
,

Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There is only one real value of ;
We find the corresponding value of using
(found by rearranging the first derivative)
There is a critical point at . We need to determine if the critical point is a maximum or minimum using
and
.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 






The real values of
are
and 
We find the corresponding value of
using
(found by rearranging the first derivative)


There are critical points at
and
. We need to determine if the critical points are maxima or minima using
and
.


At
,

Since
,
is a saddle point.
At
,


Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
The real values of are
and
We find the corresponding value of using
(found by rearranging the first derivative)
There are critical points at and
. We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




Setting each factor in the expression equal to
gives us
and 
![x=\sqrt[8]{-3*\frac{27}{-257}}=\pm \sqrt3/2](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788468/gif.latex)
The real values of
are
,
and 
We find the corresponding value of
using
(found by rearranging the first derivative)



There are critical points at
,
and
. We need to determine if the critical points are maxima or minima using
and
.


At
,

Since
,
is a saddle point.
At
,



Since
and
,
is a minimum.
At
,



Since
and
,
is a minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Setting each factor in the expression equal to gives us
and
The real values of are
,
and
We find the corresponding value of using
(found by rearranging the first derivative)
There are critical points at ,
and
. We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
At ,
Since and
,
is a minimum.
At ,
Since and
,
is a minimum.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.

The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 



Using a TI-83 or other software to find the root, we find that
,
We find the corresponding value of
using
(found by rearranging the first derivative)

There is a critical points at
. We need to determine if the critical point is a maximum or minimum using
and
.


At
,

Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Using a TI-83 or other software to find the root, we find that ,
We find the corresponding value of using
(found by rearranging the first derivative)
There is a critical points at . We need to determine if the critical point is a maximum or minimum using
and
.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are
![\frac{\partial }{\partial x} \left (\frac{\partial f }{\partial x} \right )=f_{xx}(x,y)=2\left [ sin(x)(-sin(x))+cos(x)cos(x)\right ]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788891/gif.latex)
![f_{xx}(x,y)=2\left [ -sin^2(x)+cos^2(x)\right ]=2cos^2(x)-2sin^2(x)](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788892/gif.latex)
![\frac{\partial }{\partial y} \left (\frac{\partial f }{\partial y} \right )=f_{yy}(x,y)=-2\left [ sin(y)(-sin(y))+cos(y)cos(y)\right ]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788893/gif.latex)
![f_{yy}(x,y)=-2\left [ -sin^2(y)+cos^2(y)\right ]=2sin^2(y)-2cos^2(y)](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788894/gif.latex)

To find the critical points, we will set the first derivatives equal to 






Our derivatives equal
when
and
. Every linear combination of these points is a critical point. The critical points are
,
,
, 
,
,
, 
,
,
, 
,
,
, 
We need to determine if the critical point is a maximum or minimum using
and
.



,
,
, 


Saddle point



minimum



minimum


Saddle point
,
,
, 



maximum


saddle point


saddle point



maximum
,
,
, 



maximum


saddle point


saddle point



maximum
,
,
, 


saddle point



minimum



minimum


saddle point
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Our derivatives equal when
and
. Every linear combination of these points is a critical point. The critical points are
,
,
,
,
,
,
,
,
,
,
,
,
We need to determine if the critical point is a maximum or minimum using and
.
,
,
,
Saddle point
minimum
minimum
Saddle point
,
,
,
maximum
saddle point
saddle point
maximum
,
,
,
maximum
saddle point
saddle point
maximum
,
,
,
saddle point
minimum
minimum
saddle point
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 


There is a critical point at
. We need to determine if the critical point is a maximum or minimum using
and
.


At
,


Since
and
, then there is a relative minimum at
.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There is a critical point at . We need to determine if the critical point is a maximum or minimum using
and
.
At ,
Since and
, then there is a relative minimum at
.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.

The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 


The critical points are
and
. We need to determine if the critical point is a maximum or minimum using
and
.



At
,

Since
,
is a saddle point.
At
,

Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
The critical points are and
. We need to determine if the critical point is a maximum or minimum using
and
.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are









To find the critical points, we will set the first derivatives equal to 


The exponential part of each expression cannot equal
, so each derivative is
only when
and
. That is
and
.
The critical points are
and
. We need to determine if the critical points are maxima or minima using
and
.



At
,




Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
The exponential part of each expression cannot equal , so each derivative is
only when
and
. That is
and
.
The critical points are and
. We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are







To find the critical points, we will set the first derivatives equal to 


The exponential part of each expression cannot equal
, so each derivative is
only when
and
. That is
and
.
The critical point is
. We need to determine if the critical points are maxima or minima using
and
.



^2](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/789632/gif.latex)

At
,


Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
The exponential part of each expression cannot equal , so each derivative is
only when
and
. That is
and
.
The critical point is . We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find and classify all the critical points for
.
Find and classify all the critical points for .
First thing we need to do is take partial derivatives.





Now we want to find critical points, we do this by setting the partial derivative in respect to x equal to zero.

Now we want to plug in these values into the partial derivative in respect to y and set it equal to zero.




Lets summarize the critical points:
If 

If 

Now we need to classify these points, we do this by creating a general formula
.
, where
, is a critical point.
If
and
, then there is a relative minimum at 
If
and
, then there is a relative maximum at 
If
, there is a saddle point at 
If
then the point
may be a relative minimum, relative maximum or a saddle point.

Now we plug in the critical values into
.



Since
and
,
is a relative minimum.


Since
,
is a saddle point.


Since
,
is a saddle point


Since
,
is a saddle point
First thing we need to do is take partial derivatives.
Now we want to find critical points, we do this by setting the partial derivative in respect to x equal to zero.
Now we want to plug in these values into the partial derivative in respect to y and set it equal to zero.
Lets summarize the critical points:
If
If
Now we need to classify these points, we do this by creating a general formula .
, where
, is a critical point.
If and
, then there is a relative minimum at
If and
, then there is a relative maximum at
If , there is a saddle point at
If then the point
may be a relative minimum, relative maximum or a saddle point.
Now we plug in the critical values into .
Since and
,
is a relative minimum.
Since ,
is a saddle point.
Since ,
is a saddle point
Since ,
is a saddle point
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 







There is only one critical point and it is at
. We need to determine if this critical point is a maximum or minimum using
and
.



Since
and
,
is a relative minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There is only one critical point and it is at . We need to determine if this critical point is a maximum or minimum using
and
.
Since and
,
is a relative minimum.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




There are two possible values of
,
and
.
We find the corresponding values of
using
(found by rearranging the first derivative)

![y=\frac{3\left ( \frac{\sqrt[3]{2}}{3} \right )^2}{2} =\frac{3\left ( \frac{\sqrt[3]{4}}{9} \right )}{2} =\frac{\sqrt[3]{4}}{6}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784127/gif.latex)
There are critical points at
and
. We need to determine if the critical points are maximums or minimums using
and
.


At
,

Since
,
is a saddle point.
At
,
![D(x,y)=144\left ( \frac{\sqrt[3]{2}}{3} \right )\left ( \frac{\sqrt[3]{4}}{6} \right )-4=144\left ( \frac{\sqrt[3]{8}}{18} \right )-4](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784132/gif.latex)

![f_{xx}(x,y)=6\left ( \frac{\sqrt[3]{2}}{3} \right )=2\sqrt[3]{2}>0](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/784134/gif.latex)
Since
and
,
is a relative minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There are two possible values of ,
and
.
We find the corresponding values of using
(found by rearranging the first derivative)
There are critical points at and
. We need to determine if the critical points are maximums or minimums using
and
.
At ,
Since ,
is a saddle point.
At ,
Since and
,
is a relative minimum.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




Squaring both sides of the equation gives us

Multiplying both sides of the equation by
gives us

There are three possible values of
;
,
and
.
We find the corresponding values of
using
(found by rearranging the first derivative)




There are critical points at
,
and
. We need to determine if the critical points are maximums or minimums using
and
.



At
,

Since
,
is a saddle point.
At
,


Since
,
is a saddle point.
At
,



Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Squaring both sides of the equation gives us
Multiplying both sides of the equation by gives us
There are three possible values of ;
,
and
.
We find the corresponding values of using
(found by rearranging the first derivative)
There are critical points at ,
and
. We need to determine if the critical points are maximums or minimums using
and
.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 











There is only one real value of
; 
We find the corresponding value of
using
(found by rearranging the first derivative)

There is a critical point at
. We need to determine if the critical point is a maximum or minimum using
and
.



At
,

Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
There is only one real value of ;
We find the corresponding value of using
(found by rearranging the first derivative)
There is a critical point at . We need to determine if the critical point is a maximum or minimum using
and
.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 






The real values of
are
and 
We find the corresponding value of
using
(found by rearranging the first derivative)


There are critical points at
and
. We need to determine if the critical points are maxima or minima using
and
.


At
,

Since
,
is a saddle point.
At
,


Since
,
is a saddle point.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
The real values of are
and
We find the corresponding value of using
(found by rearranging the first derivative)
There are critical points at and
. We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
At ,
Since ,
is a saddle point.
Compare your answer with the correct one above
Find the relative maxima and minima of
.
Find the relative maxima and minima of .
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.
If
and
, then there is a relative minimum at this point.
If
and
, then there is a relative maximum at this point.
If
, then this point is a saddle point.
If
, then this point cannot be classified.
The first order partial derivatives are


The second order partial derivatives are



To find the critical points, we will set the first derivatives equal to 




Setting each factor in the expression equal to
gives us
and 
![x=\sqrt[8]{-3*\frac{27}{-257}}=\pm \sqrt3/2](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/788468/gif.latex)
The real values of
are
,
and 
We find the corresponding value of
using
(found by rearranging the first derivative)



There are critical points at
,
and
. We need to determine if the critical points are maxima or minima using
and
.


At
,

Since
,
is a saddle point.
At
,



Since
and
,
is a minimum.
At
,



Since
and
,
is a minimum.
To find the relative maxima and minima, we must find all the first order and second order partial derivatives. We will use the first order partial derivative to find the critical points, then use the equation
to classify the critical points.
If and
, then there is a relative minimum at this point.
If and
, then there is a relative maximum at this point.
If , then this point is a saddle point.
If , then this point cannot be classified.
The first order partial derivatives are
The second order partial derivatives are
To find the critical points, we will set the first derivatives equal to
Setting each factor in the expression equal to gives us
and
The real values of are
,
and
We find the corresponding value of using
(found by rearranging the first derivative)
There are critical points at ,
and
. We need to determine if the critical points are maxima or minima using
and
.
At ,
Since ,
is a saddle point.
At ,
Since and
,
is a minimum.
At ,
Since and
,
is a minimum.
Compare your answer with the correct one above