Projects - Cengage Learning

Appendix C C-35
2. Use part (b) of the theorem to solve the following problems.
(a) After using a graphing utility to estimate the minimum value of the
function f(x) x 2 1
for x 0, find the exact value for this minimum.
x
Hint: Write x 2 1
as x 2 1 1
x 2x 2x .
(b) Use a graphing utility to estimate the minimum value of the function
g(x) (2x 3 3)x for x 0. Then find the exact minimum. (Adapt the
hint in the previous exercise.)
(c) A box with a square base and no top is to be constructed. The volume is
to be 27 ft 3 . Find the dimensions of the box so that the surface area is a
minimum.
(d) In Exercise 38 in Section 4.7,* a graphing utility is used to estimate the
minimum possible surface area for a circular cylinder with a volume
1000 cm 3 . Find the exact value for this minimum. Also confirm the following
fact: For the values of r and h that yield the minimum surface
area, we have 2r h.
3. (a) Proof for part (a) of the theorem: Verify that the following simple
identity holds for all real numbers a and b.
(a b) 2 4ab (a b) 2
Then use this identity to explain why part (a) of the theorem is valid.
Suggestion: First discuss the reasoning within a group; then, on your
own, write a paragraph carefully explaining the reasoning in your own
words.
(b) Proof for part (b) of the theorem: Use the result in Exercise 44 of
Section 2.3* to explain why part (b) of the theorem is valid.
*Precalculus: A Problems-Oriented Approach, 7th edition