Math205c Spring Term 2003

Midterm Examination

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You have 50 minutes to solve the problems. Good luck!

1. Let f ∈ C^{1}(D, ℝ) for some convex D ⊂^{o} ℝ^{n}. Show that

f(y) ≥ f(x) + ∇f(x) · (y − x) , x, y ∈ D

if f is assumed to be convex.

2. Classify the critical points of f : ℝ^{2} → ℝ given by

f(x, y) = x^{3} − y^{3} + 3αxy , (x, y) ∈ ℝ^{2} .

3. Let f ∈ C^{1}(D, ℝ^{n}) for some D ⊂^{o} ℝ^{n} and assume that Df(x) ∈ GL_{n}(ℝ) for all x ∈ D. Conclude that |f| has no maximum in D.

4. Consider the equation

e^{z}(x^{2} + y^{2} + z^{2}) − \sqrt{1 − z^{2}} + y = 0

and determine whether it can be solved for z about (1, 0, 0). If this is the case, give the first order Taylor expansion of z about (1, 0).

5. Let A ∈ GL_{n}(ℝ) and B ∈ ℝ^{n×n}. Compute ∂_{B}f(A) for

f : GL_{n}(ℝ) → GL_{n}(ℝ) , M ↦ M^{−1} .

[Hint: Neumann series.]