maximum and minimum

Example 1. y =
ln x (same function as in last lecture)
x
x0=e
1/e
Figure 1: Graph of y =
ln x
.
x
1
• What is the maximum value? Answer: y = .
e
• Where (or at what point) is the maximum achieved? Answer: x = e. (See Fig. 1).)
Beware: Some people will ask “What is the maximum?”. The answer is not e. You will get so used
to finding the critical point x = e, the main calculus step, that you will forget to find the maximum
1 1
value y = . Both the critical point x = e and critical value y = are important. Together, they
e e
1
form the point of the graph (e, ) where it turns around.
e
Example 2. Find the max and the min of the function in Fig. 2
Answer: If you’ve already graphed the function, it’s obvious where the maximum and minimum
values are. The point is to find the maximum and minimum without sketching the whole graph.
Idea: Look for the max and min among the critical points and endpoints.You can see from Fig. 2
that we only need to compare the heights or y-values corresponding to endpoints and critical points.
(Watch out for discontinuities!)
1
max
min
Figure 2: Search for max and min among critical points and endpoints
Example 3. Find the open-topped can with the least surface area enclosing a fixed volume, V.
r
h
Figure 3: Open-topped can.
1. Draw the picture.
2. Figure out what variables to use. (In this case, r, h, V and surface area, S.)
3. Figure out what the constraints are in the problem, and express them using a formula. In this
example, the constraint is
V = ?r2h = constant
We’re also looking for the surface area. So we need the formula for that, too:
S = ?r2 + (2?r)h
Now, in symbols, the problem is to minimize S with V constant.
2
Lecture 11 18.01 Fall 2006