Finding Maximums and Minimums
Endpoint Extremes
So far we have been discussing finding extreme values of functions over the entire real number line or on an open interval, but, in practice, we may need to find the extreme of a function over some closed interval []. If the extreme value of occurs at between and , then the previous reasoning and results still apply: either or is not differentiable at .
On a closed interval, however, there is one more possibility: an extreme can occur at an endpoint of the closed interval (Fig. 7), at or .
Fig. 7
Practice 4: List all of the local extremes of the function in Fig. 8 on the interval and state whether (i) or (ii) is not differentiable
at a or (iii) a is an endpoint.
Fig. 8
Example 3: Find the extreme values of for .
Solution: . We need to find where (i) , (ii) is not differentiable, and (iii) the endpoints.
Altogether we have four points in the interval to examine, and any extreme values of can only occur when is one of those four points: , and . The minimum of on is when , and the maximum of on is when .Sometimes the function we need to maximize or minimize is more complicated, but the same methods work.
Example 4: Find the extreme values of for .
Solution: This function comes from an application we will examine in section 3.5. The only possible locations of extremes are where or is undefined or where is an endpoint of the interval .
To determine where , we need to set the derivative equal to and solve for .
Then so , and the only point in the interval where is at .
Putting into the original equation for gives .
We can evaluate the formula for for any value of , so the derivative is always defined. Finally, the interval has two endpoints, and . and .
The maximum of on must occur at one of the points , and , and the minimum must occur at one of these three points.
The maximum value of is at , and the minimum value of is at . The graph of is shown in Fig. 9.
Fig. 9