This is mathematics year three question, please help, thanks a lot

Question 1 (34 marks) Consider the following constrained minimization model. minimize f(x) = 10:43? + 303:? + 31:; + 6:171:52 5:173 + 60:31 - 4.1:; + 8 subject to 3:31 m2 12 = O 631 +53 10 =0. (a) Write down the Lagrangian function for this model and use it to determine the firstorder necessary conditions for a constrainted local minimizer' [10] (b) Show that al 2 [3, \"3, ~8]T and a2 = [4,0, 14]T satisfy the first~order necessary condi tions, and determine the corresponding Lagrange multipliers. [8] (c) Show that (12 is a constrainted local minimizer and a1 is a constrained local maximizer. [12] (d) Estimate the change in the value of f at the local minimizer a2 if the constant in the first constraint changes from 12 to 12.1. [4]