Math205c Spring Term 2003

Assignment 19

1. Consider the function f : ℝ^{2} → ℝ given by

f(x, y) = \begin{cases}\frac{xy(x^{2}−y^{2})}{x^{2}+y^{2}} , & (x, y) ≠ (0, 0) \\ 0 , & (x, y) = (0, 0) \end{cases}

and compute its mixed second derivatives.

2. Denote ℝ^{n}\{0} by ℝ̇^{n}. A function f : ℝ̇^{n} → ℝ is called homogeneous of degree k if

f(tx) = t^{k}f(x) , t > 0 , x ∈ ℝ̇^{n} .

Show that

∇f(x) · x = kf(x)

if f is differentiable.

3. Let f ∈ C^{1}(D, ℝ) for some convex D ⊂^{o} ℝ^{n} and x, y ∈ D. Show that there exists ξ ∈ D such that

f(y) − f(x) = ∇f(ξ) · (y − x) .

4. Show that the existence of all partial derivatives for f : ℝ^{n} → ℝ at a point x ∈ ℝ^{n} does not imply its differentiability there.

5. You ask a question.

The Homework is due Friday, April 25.