170 Solutions and Hints for Selected Exercises This implies that f is differentiable at a and f ′ (a)= 0. (c) We can verify that the function f (x)= |x| satisfes the requirement. From this problem, we see that it is only interesting to consider the class of functions that satisfy (4.7) when α ≤ 1. It is an exercise to show that the function f (x)= |x|1/2 satisfes this condition with ℓ = 1 and α = 1/2. Exercise 4.3.3. Defne the function h(x)= g(x) − f (x). Then h′ (x)= g ′ (x) − f ′ (x) ≥ 0 for all x ∈ [x0,∞). Thus, h is monotone increasing on this interval. It follows that h(x) ≥ h(x0)= g(x0) − f (x0)= 0 for all x ≥ x0. Therefore, g(x) ≥ f (x) for all x ≥ x0. Exercise 4.3.5. Apply the mean value theorem twice. Exercise 4.3.6. Use proof by contradiction. SECTION 4.4 Exercise 4.4.5. Suppose that n P(x)= a0 + a1x + ··· + anx . Then apply L’Hospital’s rule repeatedly. Exercise 4.4.6. We frst consider the case where n = 1 to get ideas for solving this problem in the general case. From the standard derivative theorems we get that the function is differentiable at any x ̸ = 0 with − 1 2 − 1 −3 2 2 f ′ (x)= 2x e x = x3 e x . Consider the limit − 1 f (x) − f (0) e x2 lim = lim . x→0 x − 0 x→0 x Letting t = 1/x and applying L’Hospital rule yields − 1 x2 e t 1 lim = lim = lim = 0. t→∞ et 2 x→0+ x t→∞ 2tet 2 Similarly, − 1 x2 e lim = 0. x→0− x
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