ifiotdedepleamning di datsmachine vrartificialintelligencedx Optimization problems are related to minimizing a function (usually termed loss, cost or error function) or maximizing a function (such as the likelihood) with respect to some variable x. The Karush-Kuhn-Tucker (KKT) conditions are first-order conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. In this question, you will be solving the following optimization problem: maxx,ys.t.f(x,y)=xy8yg1(x,y)=2x2+y220g2(x,y)=x1 (a) Write the Lagrange function for the maximization problem. Now change the maximum function to a minimum function (i.e. minx,yf(x,y)=xy8y ) and provide the Lagrange function for the minimization problem with the same constraints g1 and g2.[2 pts] NOTE: The minimization problem is only for part (a). (b) List the names of all 4 groups of KKT conditions and their corresponding mathematical equations or inequalities for this specific maximization problem [2pts] (c) Solve for 4 possibilities formed by each constraint being active or inactive. Do not forget to check the inactive constraints for each point. Candidate points must satisfy the inactive constraints. [7pts] (d) List the candidate point(s) (there is at least 1). Please round answers to 3 decimal points and use that answer for calculations in further parts. [2pts] (e) Find the one candidate point for which f(x,y) is largest. Check if L(x,y) is concave or convex at this point by using the Hessian in the second partial derivative test. [2pts] (f) BONUS FOR ALL: Make a 3D plot of the objective function f(x,y) and constraints g1 and g2 using Math3d. Mark the maximum candidate point and include a screenshot of your plot. Briefly explain why your plot makes sense in one sentence. [4pts] NOTE: Use an explicit surface for the objective function, implicit surfaces for the constraints, and a point for the maximum candidate point