Read this section to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1-5.
One way to narrow our search for a maximum value of a function is to eliminate those values of which, for some reason, cannot possibly make maximum.
Proof: Assume that . By definition, , so and the right and left limits are both positive: and .
Since the right limit, , is positive, there are values of so .
Multiplying each side of this last inequality by the positive , we have and so is not a maximum.
Since the left limit, , is positive, there are values of so .
Multiplying each side of the last inequality by the negative , we have that and so is not a minimum.'
The proof for the case is similar.
When we evaluate the derivative of a function at a point , there are only four possible outcomes: or is undefined. If we are looking for extreme values of , then we can eliminate those points at which is positive or negative, and only two possibilities remain: or is undefined.Example 1: Find the local extremes of for all values of .
Solution: An extreme value of can occur only where or where is not differentiable. so only at and is a polynomial, so is differentiable for all .
The only possible locations of local extremes of are at and . We don't know yet whether or is a local extreme of , but we can be certain that no other point is a local extreme. The graph of (Fig. 4 ) shows that is a local maximum and is a local minimum. This function does not have a global maximum or minimum.
Practice 2: Find the local extremes of and .
It is important to recognize that the conditions " " or " not differentiable at a " do not guarantee that is a local maximum or minimum. They only say that might be a local extreme or that is a candidate for being a local extreme.
Example 2: Find all local extremes of .
Solution: is differentiable for all , and . The only place where
is at , so the only candidate is the point . But if then
, so is not a local maximum. Similarly, if then
so is not a local minimum. The point is the only candidate to be a local extreme of , and this candidate did not turn out to be a local extreme of . The function does not have any local extremes. (Fig. 5 )
Fig. 5
If or is not differentiable at
then the point is a candidate to be a local extreme and may or may not be a local extreme.
Practice 3: Sketch the graph of a differentiable function which satisfies the conditions:
(i) and ,
(ii) and ,
(iii) the only local maximums of are at and , and the only local minimum is at .
Once we have found the candidates for extreme points of , we still have the problem of determining whether the point is a maximum, a minimum or neither.
One method is to graph (or have your calculator graph) the function near a, and then draw your conclusion from the graph. All of the graphs in Fig. 6 have , and, on each of the graphs, either equals or is undefined. It is clear from the graphs that the point is a local maximum in (a) and (d), is a local minimum in (b) and (e), and is not a local extreme in (c) and (f).
Fig. 6
In sections 3.3 and 3.4, we will investigate how information about the first and second derivatives of can help determine whether the candidate is a maximum, a minimum, or neither.