The points of inflection of a given function are the values at which the second derivative of the function are equal to zero.

The first derivative of the function is 

, and the derivative of this function (the second derivative of the original function), is 

.

Both derivatives were found using the power rule 

. 

Solving , .

To verify that this point is a true inflection point we need to plug in a value that is less than the point and one that is greater than the point into the second derivative. If there is a sign change between the two numbers than the point in question is an inflection point.

Lets plug in , 

.

Now plug in 

.

Therefore,  is the only point of inflection of the function.

In order to find all the points of inflection, we first find  using the power rule twice, .

Now we set .

.

Now we factor the left hand side.

From this, we see that there is one point of inflection at .

For the point of inflection, lets solve for x for the equation inside the parentheses. 

 In order to find all the points of inflection, we first find  using the power rule twice .

Now we set .

Thus the points of inflection are   and 

Which of the following is a point of inflection on f(x)?

To find points of inflection, we need to find where the second derivative is 0.

So, find f''(x)

So, we have a point of inflection at x=0.

Find f(0) to find the y-coordinate:

So our point is at .

Which of the following is a point of inflection of f(x) on the interval ?

To find points of inflection, we need to know where the second derivative of the function is equal to zero. So, find the second derivative:

So, where on the given interval does ?

Well, we know from our unit circle that ,

So we would have a point of inflection at , but we still need to find the y-coordinate of our POI. find this by finding 

So our POI is:

For a graph to have an inflection point, the second derivative must be equal to zero. We also want the concavity to change at that point. 

, for all real numbers, but this is a line and has no concavity associated with it, so not this one.

, there are no real values of  for which this equals zero, so no inflection points.

, same story here.

, so no inflection points here.

This leaves us with 

, whose derivatives are a bit more difficult to take.

, so by the chain rule we get

So,  when . So 

.  This is our correct answer.

To find the inflection point we must find where the second derivative of a function is 0.

Calculating the second derivative is fairly simple. We just need to know that :

The first derivative is 

and if we take the derivative once more we get 

.

Setting this equal to zero, we get 

And now all we have to do is plug this value, 3/5, into our original polynomial, to get the answer.

Points of inflection occur when the second derivative changes signs.

The second derivative equals zero at and . However, the factor  has degree two in the second derivative.  This indicates that is a root of the second derivative with multiplicity two, so the second derivative does not change signs at this value.  It only changes signs at .  Since , the point of inflection is .

To find the inflection points of a function we need to take the second derivative and find which values make it zero.

To find the first and second derivative we will need to apply the power rule, .

Given, 

and applying the power rule we get,

.

Points of inflection happen when the second derivative changes signs.  The quadratic above changes signs at and . 

This can be determined by factoring,

,

or by the quadratic formula,

.

The point of inflection can be found by setting the second derivative equal to 0.

The power rule is given by:

Use the power rule twice to find the second derivative.

Set the second derivative equal to  and solve for .

Find the point of inflection by plugging  back into the original equation.

Therefore, the point of inflection is