Math 104 homework 9 consider the case when fx x5 how does

Question

Math 104: Homework 9-

1. Suppose that f_n is a sequence of real-valued functions defined on an interval [a, b] that converges uniformly to a function f. Let x₀ ∈ [a, b], and suppose that lim_{x→x_0}fn(x) = l_n, for all n ∈ N.

(a) Prove that l_n is a Cauchy sequence, and hence that it converges to a limit l.

(b) Prove that lim_{x→x_0}f(x) = l.

2. (a) Let f: R → R be a twice differentiable function, where f(0) = 0 and f''(x) ≥ 0 for all x > 0. Prove that f(x)/x is increasing for x > 0.

(b) If f: R → R is twice differentiable, f(0) = 0 and if f(x)/x is increasing for all x > 0, show that f''(x) ≥ 0 for some x > 0, but not necessarily for all x > 0.

3. Let f(x) = |x|³. Compute f'(x), f''(x), and show that f⁽³⁾(0) does not exist.

4. Suppose that f is differentiable at a point a. Define

L₁(a, h) = (f(a + h) - f(a - h)/2h),

L₂(a, h) = (-f(a + 2h) + 8 f(a + h) - 8 f(a - h) + f(a - 2h)/12h).

(a) Prove that lim_h→0 L_i(a, h) = f'(a) for i = 1, 2.

(b) Consider the case when f(x) = x⁵. How does |L_i(a, h) - f'(a)| behave as h → 0 for i = 1, 2? Is there a difference in the rate of convergence?