To find the inflection points and concavity of any function, we must first find its second derivative. If we set the equation for the second derivative equal to 0, we can then solve for the x values of our function’s inflection points:

Because the function is a polynomial, we know it is continuous everywhere, and therefore we know our two inflection points are valid. To find the intervals on which the funciton is concave up and concave down, we simply plug in a value below our first inflection points, a value between our inflection points, and a value above our second inflection point into the equation for f ’’(x) and see if we get a negative or positive value. Negative indicates the function is concave down on the interval, while positive indicates the function is concave up on the interval.

Plugging in arbitrary values below, between, and above our inflection points, we can see the second derivative is negative (the funciton is concave down) on (0,6), and positive (the function is concave up) on (-∞,0) and (6,∞).

Using the method of substitution will give an indeterminate form of .

Therefore, it is possible to use L'Hopital to determine the limit.

Write the formula for L'Hopital.

Take the derivative of the numerator and denominator of .

Since substitution will yield an indeterminate form, apply L'Hopital again.

The limit is  as .

To determine the derivative of this function: , it is necessary to use two chain rules after taking the derivatives of tangent and natural log functions.

Use brackets to identify the terms which require the chain rule.

 represents the rate of change of  when  In particular, this represents the change in cost of renting the truck when it is driven  miles. This change gives the approximate increase in cost if the truck is driven an additional mile. Notice that the units of the derivative are the units of  over the units of . Therefore the interpretation of this term must have units.

To understand what's happening with the velocity given the acceleration curve, integrate the acceleration function to obtain the velocity curve.

Set the velocity equal to zero and solve for time.

Therefore, the velocity must be zero when .  All of the other statements are false.

Concavity describes where on the original function the curve concaves up or down. Algebraically, concavity is the second derivative of the original function.

Set the concavity function equal to zero.

Test the values to the left and right of . Anywhere to the left of  will yield negative concavity, and the right of  will yield positive concavity. The original equation does not necessarily always concave up.

Take the antiderivative of the concavity function to obtain the derivative function, or the slope of the original function.

Since the derivative function has an  term, the slopes the original function are not necessarily constant values.  

Set the derivative function equal to zero and solve for .

The slope of the original function must have zero slopes at the roots .

Therefore, the only true statement is:

To find the limit of a function at a particular value we first want to simplify our function .

We can factor the numerator which results in the following.

This expression can be simplified to

.

Then we plug in  to get the final answer .

When finding the limit of a function at a particular point, you first want to see if direct substitution of the point results in an answer.

For the function,

if we plug in 0, we will get  , which is undefined. Since direct substitution did not succeed we can use L'Hospital's Rule. 

L'Hospital's Rule states that if,   

  or  ,  where  is any real number then, 

.

Let , and .

Now we find  and .

.

We can rewrite our limit as: 

Evaluating the limit gives us

.

If we plug in  , we will get  .

So we can use L'Hospital's Rule.

L'Hospital's Rule states that if  

  or  ,  where  is any real number then,

.

Let  , and  .

We now find , and .

Now we can rewrite our limit as:

After plugging in   , we get

If we plug in 0, we get  

So we can use L'Hospital's Rule. 

L'Hospital's Rule states that if  

  or  ,  where  is any real number then, 

.

Set , and  . Then find  and .

,  .

Now we can rewrite our limit as:

Now if we plug in 0, we get,