MATH 163, Spring 2001 Due Date: Name(s):
Extra Project 4.1: Maximum and Minimum Values on a Closed Interval
Objective
To illustrate how Maple can be used to find the maximum and minimum values of a function on a closed
interval.
Narrative
If you have not already done so, read Section 4.1 of the text. In particular, be sure to read Example 4,
p. 224: below we use this example to illustrate how Maple can be used to find the maximum and minimum
values of f(x) = 3x4
- 16x3
+ 18x2
on the closed interval [-1, 4].
Task
a) Type the command lines in the left-hand column below into Maple in the order in which they are listed.
The effect of each command is described in the right-hand column for your reference.
> # Project 4.1: Maximum and Minimum Values on a Closed Interval
> restart; Clear Maple's memory.
> f := x -> 3*x^4-16*x^3+18*x^2; Let f(x) = 3x4
- 16x3
+ 18x2
.
> plot(f(x),x=-1..4); Plot the graph of f over the interval [-1, 4].
> f1 := D(f); Let f1 denote the first derivative f of f.
> cn:= fsolve(f1(x)=0,x,x=-1..4); Find the critical numbers of f on [-1, 4] by solving
the equation f (x) = 0.
> print(cn[1],f(cn[1])); The first critical number and the value of f there.
> print(cn[2],f(cn[2])); The second critical number and the value of f
there.
> print(cn[3],f(cn[3])); The third critical number and the value of f there.
> print(-1,f(-1)); The left-hand endpoint of [-1, 4] and the value of
f there.
> print(4,f(4)); The right-hand endpoint of [-1, 4] and the value
of f there.
> plot({f(x),f1(x)},x=-1..4,y=-30..40); Plot the graphs of f and f over a y-range large
enough to capture all relevant behavior.
At this point make a copy of your typed input and Maple's responses (both text and graphics). Then, ...
b) What is the maximum value of f on the closed interval [-1, 4]? What is the minimum value of f on the
closed interval [-1, 4]?
c) By hand, label the graphs of f and f in the last plot, and highlight that part of the graph of f over
which f is increasing, and that part of the graph of f over which the values of f are positive.
Comments
To find the critical numbers of f we must find the values of x in the domain of f for which f (x) = 0 and
f (x) does not exist. We restricted our attention to points at which f (x) = 0 in the above example since
f(x) is a polynomial function so f (x) exists for all x. If f(x) were a quotient, however, then we would need
to find not only the values of x for which the numerator of f (x) -- namely numer(f1(x)) -- is zero, but
also the values of x for which the denominator of f (x) -- namely denom(f1(x)) -- is zero.