CSULA Optimality Conditions for Various Functions Questions

Question Description

I’m working on a mathematics multi-part question and need guidance to help me understand better.

Please explain how to tackle this problem.

Show that the function f(x) = 8×1 + 12×2 + x21 ? 2×22 has only one stationary point, and that it is neither a local maximizer nor a local minimizer, but a saddle point (which means the Hessian has both positive and negative eigenvalues).

(b) Show that the 2-dimensional function f(x1,x2) = (x2?x21)2?x21 has only one stationary point, which is neither a local maximizer nor a local minimizer, but a saddle point.

(c) Find all the critical points of the 2-dimensional function f(x1,x2) = (x21 ? 1)2 + x22. Which are local minimizers? Which are not local minimizers?

(d) Find all the critical points of the 2-dimensional function f(x1,x2) = (x21?1)2+(x22?1)2. Which are local minimizers? Which are not local minimizers?

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