Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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