Mathematics 524 Test #1 Name:

Show your work to get credit.

1. (15 points) Let D be a subset of R^{n} and let f(x) a real valued function defined on D. Then define the following:

(a) x^{∗} is a global minimizer of f on D.

(b) x^{∗} is a strict local minimizer of f on D.

(c) Define the gradient of a function.

2. (10 points) State the second order Taylor’s theorem for functions f(x) of a vector variable x.

3. (15 points) Give explicit examples of (a) A function f with a global maximizer that is not a strict local maximizer.

(b) A function f with a saddle point.

(c) A function f(x) defined on R that has f(x) > 0 for all x, but which has no local minimizers.

4. (15 points) Let f(x, y) be a function that has a continuous partial derivatives and so that (2, −3) is a global minimizer of f(x, y). Then show that (2, −3) is a critical point of f(x, y).

5. (15 points) Let f(x, y) = x^{3} − 3x + y^{3} − 12y = 7. Find all critical point of f(x, y) and classify as to local maximizers, local minimizers, or saddle points.

6. (15 points) Let f(x) be a function on R^{n} so that the Hessian Hf(x) is positive definite for all x. Then show that any critical point of f(x) is a global minimizer.

7. (15 points) For the following matrices determine if they are positive definite, negative definite or indefinite.

(a) [\begin{matrix}1 & 2 & 3 \\ 2 & −2 & 4 \\ 3 & 4 & 7 \end{matrix}]

(b) [\begin{matrix}1 & −2 \\ −2 & 3 \end{matrix}]