PFA

PFA


Consider the function ?: ℝ2 → ℝ defined by ?(?) = − 2x1 2 + 5x1x2 − 4x2 2. (a) Find the first derivative ??(?) and the second ?2?(?) of f, and using the characterization of concave functions twice differentiable, verify that f is concave. (Hint: to show that a symmetric matrix is semi-defined negative you can use the fact that the condition equates to the matrix eigenvalues being non-positive.) (b) Consider the function �̂�: ℝ2 → ℝ4 defined by �̂�(?) = ( −?1 −?2 ?1 − 1 ?2 − 1 ) Defining ? = {? ? ℝ2: �̂�?(?) ≤ 0, ? = 1,2,3,4}, check that ? is compact and convex, and find the solution to the problem of ?????ℝ2?(?) subject to ? ? ?.