# 12.8.2. Maximum and Minimum Values

Find the local maximum or local minimum points of the function .

• AThe function has a local minimum at .
• BThe function has a local maximum at .
• CThe function has a local maximum at .
• DThe function has a local minimum at .

### Example

Find the local maximum or local minimum points of the function .

### Solution

Stationary points are where the first derivative is equal to 0. First, find using the chain rule.

Now set equal to 0 and solve for .

Now evaluate the function when .

Therefore, the function has a stationary point at .

Finally, determine whether this point is a local minimum or a local maximum by evaluating the second derivative at this point.

As , this does not show whether the stationary point is a local minimum or a local maximum. Instead, consider the slope on either side of the stationary point.

The slope changes from negative to positive, and therefore the stationary point is a local minimum.

Hence, the function has a local minimum at the point .

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