# 12.8.2. Maximum and Minimum Values

Determine the local minimum and local maximum values of the function .

• A local maximum value , local minimum value
• B local minimum value , local maximum value
• CThe function has no local minimum or local maximum values.

### Example

Determine the local minimum and local maximum values of the function .

### Solution

Given that the function is differentiable, , and , then has a local maximum value at . Likewise, if and , then has a local minimum value at . First, find the derivative of the function . Keep in mind that the function is not differentiable at . Also, when , the function can be written as , and when , the function can be written as , or . Differentiate as follows:

Now substitute 0 for in the equation and solve for .

Find the second derivative of the function , using as the first derivative.

Next, differentiate .

Substituting 0 for in the equation and solving for gives the same value of . Since this value of is less than 3, is actually the correct derivative to use. Find the second derivative of the function , using as the first derivative.

Since is always negative, the function has a local maximum value at . Next, substitute for in the equation and solve for to find the local maximum.

To check if the function has a local maximum or minimum at , determine if is positive or negative when is slightly greater than 3 and when is slightly less than 3. When is slightly greater than 3, , so is positive. When is slightly less than 3, , so is negative. This means that the function has a local minimum at . Now substitute 3 for in the equation and solve for to find the local minimum.

Thus, the function has a local maximum value of and a local minimum value of 0.

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