167 and f (x) =( x2e−1/x2 , if x̸ =0; 0, if x =0. 2. Suppose that ϕ is bounded and differentiable on R. Define the function f (x) =( xnϕ(1/x), if x̸ =0; 0, if x =0. Show that if n≥2, the function is differentiable on Rand find its derivative. Show that if n=1 and limx→∞ϕ(x) does not exists, then f is not differentiable at 0. (b) Hint: Observe that f ′ 1 2nπ =−1+c <0 and f ′ 1 (2n+1)π =1+c >0. SECTION 4.2 Exercise 4.2.1. Define the function h(x) =f (x)−g(x). Then hhas an absolute maximum at x0. Thus, h′(x0) =f ′(x0)−g′(x0) =0, which implies f ′(x0) =g′(x0). Exercise 4.2.3. The inequality holds obviously if a=b. In the case where a̸=b, the equality can be rewritten as sin(b)−sin(a) b−a ≤ 1. The quotient sin(b)−sin(a) b−a is the slope of the line connecting(a, f (a)) and(b, f (b)). We need to show that the absolute value of the slope is always bounded by 1, which can also be seen from the figure. The quotient also reminds us of applying the Mean Value Theorem for the function f (x) =sin(x). Consider the case where a<band define the function f : [a,b] →Rby f (x) =sin(x). Clearly, the function satisfies all assumptions of the Mean Value Theorem on this interval with f ′(x) =cos(x) for all x ∈(a,b). By the Mean Value Theorem, there exists c ∈(a,b) such that f (b)−f (a) b−a =f ′(c) =cos(c),
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