Download Homework 8

Homework 8
Math 15100 (section 51), Fall 2014
This homework is due in class on Wednesday, November 26th. You may cite results from class as
appropriate. Unless otherwise stated, you must provide a complete explanation for your solutions,
not simply an answer. You are encouraged to work together on these problems, but you must write
up your solutions independently.
1. Find the critical points of the following functions. Find and classify all of the extreme values.
That is, find all points c where f takes on an extreme value, classify these points as local,
absolute or endpoint maxima or minima, and give the value f (c) at each such point.
√
(a) (Ex. 4.4.1) f (x) = x + 2
(b) (Ex. 4.4.4) f (x) = 2x2 + 5x − 1 for x ∈ [−2, 0]
√
(c) (Ex. 4.4.13) f (x) = (x − x)2
√
(d) (Ex. 4.4.19) f (x) = sin2 x − 3 cos x for x ∈ [0, π]
(e) (Ex. 4.4.25) f (x) =
(f) (Ex. 4.4.28) f (x) =



−2x
0≤x<1
x−3 1≤x≤4


5 − x 4 < x ≤ 7

2


2 − 2x − x
|x − 2|


 1 (x − 2)3
3
−2 ≤ x ≤ 0
0<x<3
3≤x≤4
2. (Ex. 4.4.45) Let p and q be positive rational numbers and let f (x) = xp (1 − x)q on [0, 1]. Find
the absolute maximum value of f .
3. (Ex. 4.4.44) Show that of all rectangles with a diagonal of length c, the square has the largest
area. (Hint: Let one of the edges of the rectangle have length x, compute the length of the
other side and the area of the rectangle in terms of x. Find the value of x at which this area
is maximized.)
4. (Ex. 4.4.48) A piece of wire of length L is to be cut into two pieces, one piece to form a square
and the other piece to form an equilateral triangle. How should the wire be cut in order to:
(a) minimize the sum of the areas of the square and the triangle?
(b) maximize the sum of the areas of the square and the triangle?
(Hint: Let one of the lengths be x, and the other be L − x. What is the sum of the areas?)
1
5. Let f be an everywhere differentiable function with f (0) = 0 and |f 0 (x)| ≤ 1 for all x. Prove
that |f (x)| ≤ |x| for all x. (Hint: Mean Value Theorem)
6. (Ex. 4.2.56,57) Assume that f and g are differentiable on the interval (−c, c) with f (0) = g(0).
(a) Prove that if f 0 (x) > g 0 (x) for all x ∈ (0, c), then f (x) > g(x) for all x ∈ (0, c). (Hint:
Let h = f − g.)
(b) Prove that if f 0 (x) > g 0 (x) for all x ∈ (−c, 0), then f (x) < g(x) for all x ∈ (−c, 0).
(c) Prove that tan x > x for all x ∈ (0, π/2).
7. Let f be a function, and assume that f 0 (c) and f 00 (c) exist. Recall that f (c + h) ≈ f (c) + hf 0 (c)
for h ≈ 0.
(a) Prove that if f 00 (c) > 0 then f (c + h) > f (c) + hf 0 (c) for h sufficiently close to (but not
equal to) 0 (that is, for all h ∈ (−p, p) but h 6= 0 for some p > 0).
(a) Prove that if f 00 (c) < 0 then f (c + h) < f (c) + hf 0 (c) for h sufficiently close to (but not
equal to) 0.
(Hint: Apply the second derivative test to g(h) = f (c + h) − f (c) − hf 0 (c).)
1
h
8. If f is a function and L is a real number, prove that lim f (x) = L iff lim f
x→−∞
1
h
h→0−
9. Use the formulas lim f (x) = lim f
x→∞
h→0+
1
h
= L.
and lim f (x) = lim f
x→−∞
2x3 + x + 4
x→∞
x3 − 6
(b)
(a) lim
h→0−
lim
x→−∞
x2
to compute
x
+1
10. (Ex. 4.4.43) Let P (x) = an xn + an−1 xn−1 + · · · + a0 be a polynomial with positive leading
coefficient (i.e. an > 0). The goal of this problem is to prove that lim P (x) = ∞.
x→∞
P (x)
1
= an (Hint: lim f (x) = lim f
).
x→∞
xn
h
h→0+
(b) Conclude that for any ε > 0 there is some K > 0 with P (x) > (an − ε)xn for all x ≥ K.
(a) Prove that lim
x→∞
(c) Let ε =
x ≥ K 0.
an
2 .
Prove that for any M > 0, there is some K 0 with
an n
2 x
≥ M whenever
(d) Conclude that P (x) ≥ M whenever x ≥ max{K, K 0 }, and deduce that lim P (x) = ∞.
x→∞
2