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:

The first step to finding the slope of the line normal to a point is to find the slope of the tangent at this point.

The slope of this tangent, in turn, is found by finding the value of the derivative of the function at this point.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

Evaluating the function  at the point 

The slope of the tangent is

The slope of the normal line is the negative reciprocal of this value. Thus for this problem the normal is

Use the power rule to find this derivative. 

Remember the power rule is:

Now lets apply this to our problem.

Recall that the derivative of a constant is zero.

Thus, the derivative is .

First, use the power rule to find the derivative. 

Remember the power rule is:

Now lets apply this to our problem.

Recall that the derivative of a constant is zero.

Thus, the derivative is 

Now, substitute  for .

Use the power rule to find this derivative. 

Remember the power rule is:

Now lets apply this to our problem.

Thus, the derivative is .

First, find the derivative using the power rule. 

Remember the power rule is:

Now lets apply this to our problem.

Recall that the derivative of a constant is zero.

The derivative is .

Now, substitute  for .

Use the power rule to find this derivative. 

Remember the power rule is:

Now lets apply this to our problem.