# 12.8.2. Maximum and Minimum Values

Determine the absolute maximum and minimum values of the function over the interval .

• A The absolute maximum value is 64 at , and the absolute minimum value is 4 at .
• B The absolute maximum value is 25 at , and the absolute minimum value is 4 at .
• C The absolute maximum value is 64 at , and the absolute minimum value is 25 at .
• D The function has no absolute maximum or minimum values.

### Example

Determine the absolute maximum and minimum values of the function over the interval .

### Solution

The piecewise function is defined by one piece when and another piece when , so it is possible that is a critical point. To test this, first evaluate the right-hand derivative of at as follows:

Now evaluate the left-hand derivative of at .

Since , does not exist at . This means that a critical point of occurs at . Next, find when .

Now substitute 0 for in the equation and solve for .

Since falls outside the interval , there is no need to check this value of . Next, find when .

Now substitute 0 for in the equation and solve for .

Since falls outside the interval , there is no need to check this value of , either. This means the only values of to check are and the endpoints of the interval , which are and . First, check by substituting for in the equation .

Next, check by substituting for in the equation .

Finally, check by substituting 5 for in the equation .

Thus, the absolute maximum value of the function is 64 at , and the absolute minimum value of the function is 4 at .

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