# 12.8.2. Maximum and Minimum Values

Determine the points of local maximum and minimum in the domain of the function .

• Ais the local maximum point, and is the local minimum point.
• Bis the local minimum point, and is the local maximum point.
• Cis the local maximum point, and the function doesn't have a local minimum point.
• Dis the local minimum point, and the function doesn't have a local maximum point.

### Example

Determine the points of local maximum and minimum in the domain 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 as follows:

Now substitute 0 for in the equation and solve for .

Next, find the second derivative of the function .

Now substitute 0 for in the equation to find .

Since , the function has a local minimum at . Next, substitute 0 for in the equation and solve for to find the local minimum.

Now substitute for in the equation to find .

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

Thus, the function has a local minimum point at and a local maximum point at .

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