The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

To take these derivatives, make use of the following rules:

Set the second derivative to zero and find the values that satisfy the equation:

These can be shown to be points inflection by observing how the signs on the plot of ; note how the function crosses the x-axis at these values

Now, plug these values back in to the original function to find the values of the function that match to them:

The points of inflection are , , 

First, find the derivative using the power rule:

Remember that the power rule is:

Apply this to our problem to get

Now, substitute  for .

First, use the power rule to find the derivative.

Remember that the power rule is:

Apply this to our problem to get

Now, substitute  for .

Find the derivative using the power rule.

Remember that the power rule is:

Apply this to our problem to get

Now, substitute  for . 

First, find the derivative using the power rule. 

Remember that the power rule is:

Apply this to our problem to get

Now, substitute  for .

Use the power rule to find this derivative.

Remember that the power rule is:

Apply this to our problem to get

Thus, the derivative is .

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the values that satisfy the equation:

These points can be shown to be points of inflection by how the sign changes at points just adjacent to them:

For 

For 

Now, plug these values back in to the original function to find the values of the function that match to them:

The two points of inflection are

, 

To verify an inflection point, plug a x value higher and lower into the second derivative to find if there is a sign change. If a sign change occurs around the inflection point than it is in fact a true inflection point. To check , plug in a value higher and a value lower than it.

Use the product rule to find the derivative.

Remember that the product rule is:

Apply this to our problem to get

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the values that satisfy the equation:

This can be shown to be a point of inflection by how the sign changes for  on either side of it:

Now, plug this value back in to the original function to find the value of the function that matches:

The point of inflection is .

To verify this is a true inflection point, plug in x values that are higher and lower than four. If a sign change occurs around four than it is in fact an inflection point.

The points of inflection of a function occur where the second derivative of the funtion is equal to zero.

Find this second derivative by taking the derivative of the function twice:

Set the second derivative to zero and find the values that satisfy the equation:

The only value of  which satisfies the equation is  

This can be shown to be a point of inflection due to  having opposite signs on either side of it:

Now, plug these values back in to the original function to find the values of the function that match to them:

The point of inflection is .

It can be confirmed that  is a point of inflection due to the sign change around this point. Picking a greater and lower value , observe the difference in sign of the second derivative: