Mathematics 524 Test #2 Name:

Show your work to get credit.

1. (30 points) Carefully state or define the following: (a) C is a convex subset of R^{n}.

(b) x is a convex combination of x_{1}, x_{2}, …, x_{k}.

(c) The second order Taylor’s Theorem for functions of several variables.

(d) The Arithmetic-Geometric mean inequality.

(e) A posynomial.

(f) The basic Geometric Program (GP).

2. (20 points) Let C be a convex subset of R^{n}.

(a) Let f be a real valued function defined on C. Then define what it means for f to be a convex function on C.

(b) If f_{1} and f_{2} be convex functions on C. Then show the sum f_{1} +f_{2} is also a convex function on C.

3. (15 points) Let f : R^{n} → R be a function so that the Hessian Hf is positive semi-definite at all points. Then show that any critical point of f is a global minimizer.

4. (15 points) Find the maximum and the minimizer of xyz subject to the constraints 2x + 3y + 4z = 96, x, y, y > 0.

5. (20 points) Find the minimum and minimizer of f(x, y) = x + 2y + \frac{1}{xy^{2}} .