Math205a Fall Term 2002

Assignment 7

1. Let f : D → ℝ be differentiable at x_{0} ∈ D ⊂^{o} ℝ. Show that f is Lipschitz continuous at x_{0}.

2. Analyze the continuity and differentiability of the function

f(x) := \begin{cases}0 , & x ≤ 0 , \\ \frac{1}{x}e^{− \frac{1}{x^{2}}} , & x > 0 . \end{cases}

What can you say about higher order derivatives at x = 0?

3. Let f : D → ℝ be differentiable at x_{0} ∈ D ⊂^{o} ℝ. Prove or disprove: (i) f is strictly increasing at x_{0} implies that f′(x_{0}) > 0. (ii) f′(x_{0}) = 0 implies that f has a local minimum or maximum at x_{0}. (iii)f is Lipschitz continuous at x_{0} and has local minimum there implies that f is differentiable at x_{0}.

4. Let a, b ∈ ℝ with a < b and f : [a, b] → ℝ. Define

d^{+}f(x_{0}) = lim_{x→x_{0}+0} \frac{f(x) − f(x_{0})}{x − x_{0}} , x_{0} ∈ [a, b) ,
d^{−}f(x_{0}) = lim_{x→x_{0}−0} \frac{f(x) − f(x_{0})}{x − x_{0}} , x_{0} ∈ (a, b] ,

the right and left derivative of f at x_{0}, respectively, if the limit exists.

(i) Show that f is differentiable at x_{0} ∈ (a, b) if and only if d^{±}f(x_{0}) exist and coincide. (ii) Assume that f takes on a minimum or a maximum at a or b and is one-side differentiable there. What can you say about d^{+}f(a) and d^{−}f(b)?

The Homework is due Monday, Novermber 25