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:

It can be shown that this is a point of inflection by the change in sign of the second derivative with points before and after this value

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

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:

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.