Consider the following function
(a) Is f coercive? Justify your answer.
(b) Identify all local minimizers, saddle points, and global minimizers of f.
(c) Using Matlab’s surf and plot3 commands, plot f and its stationary points in the interval −5 ≤ x1 ≤ 5
and −5 ≤ x2 ≤ 5. Submit the printout of the plot with your responses.
(a) Let A and B be two positive definite matrices. Determine the positive definiteness of the following
(i) A2; (ii) A + B; (iii) AB; (iv) AAT; (v) ABA:
(b) Classify the following three matrices according to whether they are positive or negative definite,
semidefinite, or indefinite
Let f 2 C2(Rn) and consider two fixed vectors x 2 Rn and u 2 Rn. We define a function
’(γ) := f(x + γu) of a single variable γ 2 R.
(a) Compute the derivatives ’0(γ) and ’00(γ).
(b) What does the quantity ’0(0) measure regarding f(x)?