5.1. OPTIMIZATION WITH DERIVATIVES 183
We know r > 0. Looking at the sign chart for the derivative, plugging in r D 2 and r D 4, we
get:
.0/ .3:17/ C C C .1/
so the critical value x D 3:17 is a minimum. Plugging the r value into the formula for h, we get
that
h
Š
6:34
. Our answer is
r
D
3:17
cm and
h
D
6:34
cm.
˙
Let’s see how the second derivative test shakes out in the previous example.
A
00
D 4 C
800
r
3
which is positive for any r > 0. So the curve is concave up in the possible region, and our
critical value is a minimum; the second derivative test agrees with the sign chart for the first
derivative test.
When you are working optimization story problems, it is critical to make sure the values you get
make sense. Negative lengths, for example, probably mean you made a mistake. ese problems
also have the property that some of what you did may be needed again in a later step. Using a
neat layout, possibly informed by the steps given in Knowledge Box 5.5, will help.
PROBLEMS
Problem 5.9 For each of the following functions, find the global maximum and minimum of
the function, if they exist, on the stated interval.
1. f .x/ D 2x 1 on .2; 2/
2. g.x/ D 3x C 1 on Œ3; 1
3. h.x/ D x
2
C 4x C 3 on Œ1; 4
4. r.x/ D ln.x/ on Œ1; e
3
5. s.x/ D e
x
2
, on Œ2; 3
6. q.x/ D x
3
16x C 1, on Œ4; 4
184 5. OPTIMIZATION
Problem 5.10 How many different horizontal tangent lines do the following functions have?
Be sure to justify your answer.
1. f .x/ D cos.x/
2. g.x/ D xe
x
3. h.x/ D x.x 5/.x C 5/
4. r.x/ D cos.x/ C x
5. s.x/ D ln.x
4
C 4x
3
C 5/
6. q.x/ D tan
1
.x/
Problem 5.11 Construct a function that has exactly three horizontal tangent lines—all differ-
ent from one another.
Problem 5.12 What is the largest number of horizontal tangent lines that a polynomial of
degree n can have?
Problem 5.13 If y D ax
2
C bx C c with a ¤ 0, give a set of steps for finding the global op-
timum (there is exactly one), and determining the type of optima it is.
Problem 5.14 For each of the following functions, find the global maximum and minimum of
the function, if they exist, on the stated interval.
1. f .x/ D
4x
x
2
C 2
on .1; 1/ 4. r.x/ D x
2
e
x
on .0; 1/
2. g.x/ D e
x
3
5xC12
on Œ5 W 5 5. s.x/ D
p
x
2
C 1
x
, on Œ4; 1 [ Œ1; 4
3. h.x/ D xe
2x
on .0; 1/ 6. q.x/ D 25 x
4
, on Œ1; 2
Problem 5.15 Suppose for m.x/ that when a b we have that m.a/ m.b/, and assume that
m.x/ is continuous, differentiable, and not constant.
1. What do we know about m
0
.x/? Explain.
2. Prove that the critical values of m.f .x// and f .x/ are the same.
3. Is this a problem relevant to other problems in this section? Why?
5.1. OPTIMIZATION WITH DERIVATIVES 185
Problem 5.16 Find the domain of
h.x/ D
p
6 3x x
2
and find its global maximum and minimum on the domain.
Problem 5.17 If an open-topped can holds 400 cc, what radius and height minimize the
amount of material needed to make the can?
Problem 5.18 What are the ratio of the sides of a rectangle of perimeter P that maximizes
the area over all rectangles with that perimeter?
x
2 m
Problem 5.19 Suppose that we cut square corners out of a 2 m 2 m square of pasteboard and
tape up the sides to make an open-topped box. What side length x of the square maximizes the
volume of the box?
Problem 5.20 What point on the line
y D x 6
is closest to the origin?
Problem 5.21 What point on the line
y D 3x C 5
is closest to the origin?
Problem 5.22 What point on the line
y D 4x C 1
is closest to the origin?
186 5. OPTIMIZATION
Problem 5.23 If we use 240 m of fence to lay out three pens like those shown above, what
length and width for one pen maximize the area enclosed?
Problem 5.24 Suppose that the top of a can is made of a material that costs twice as much as
the bottom or sides. Find the radius and height of a can containing 200 cc that minimizes the
cost.
Problem 5.25 One side of a rectangular pen must be made of opaque material that costs three
times as much as the material used to make the other three sides. What dimensions minimize
the cost if the pen must have an area of 60 m
2
?
Problem 5.26 A rocket takes off, straight up, so that for t 0 the height of the rocket is
4t
2
m
.
A camera 40 m from the rocket tracks its takeoff.
1. Make a sketch of the situation.
2. Find an expression for the rate in rad=sec for the camera’s rate of spin.
3. When is the camera spinning fastest?
4. What is the camera’s rate of spin when it is spinning fastest?
5.1. OPTIMIZATION WITH DERIVATIVES 187
Problem 5.27 Suppose we are optimizing the area of cells of a rectangular grid like the one
shown above fixing the total length of the sides of the grids. Prove that the solution always places
the same amount of material into vertical and horizontal cell sides.
r
h
Problem 5.28 What radius and height minimize the material needed to make an open-topped
can with a volume of 600 cc?
Problem 5.29 Maximize
q.x/ D x
2
e
2x
for x 0.
Problem 5.30 What is the global maximum of the function
y D e
4x
2
Can you do this problem without calculus? Explain.
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