1. 9. Let f (x) = x 2 if x ≥ 0 and f (x) = 0 if x (a) Show that f is differentiable at x = 0. (b) Find f ′(x) for all real x and sketch the graph of f ′. (c) Is f ′ continuous on ? Is f ′...

1. 9. Let f (x) = x²
if x ≥ 0 and f (x) = 0 if x

(a) Show that f is differentiable at x = 0.

(b) Find f ′(x) for all real x and sketch the graph of f ′.

(c) Is f ′ continuous on
? Is f ′ differentiable on R?

2. Let f (x) = x²
if x is rational and f (x) = 0 if x is irrational.

(a) Prove that f is continuous at exactly one point, namely at x = 0.

(b) Prove that f is differentiable at exactly one point, namely at x = 0.