MAT2615 Assignment 3 2026 – DUE 2026 [COMPLETE ANSWERS];100% TRUSTED WORKING

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MAT2615 Assignment 3 2026 – DUE 2026 [COMPLETE ANSWERS];100% TRUSTED WORKINGS.1. (Sections 10.1, 10.2) Consider the R 2 − R function f defined by f (x, y) = x 2 − 6x + 3y 2 − y 3 . (a) Find all the critical points of f. (The function has two critical points.) (5) (b) Use Theorem 10.2.9 to determine the local extreme values and minimax values of f. Also determine (by inspection) whether any of the local extrema are global extrema. (5) [10] 2. (Sections 2.6, and Chapter 10) Let L be the line with equation y = x − 1. Find the minimum distance between L and the point (4, 5) by using (a) Theorem 10.2.4 (5) (b) The Method of Lagrange. (5) Hints. • Minimize the square of the distance between the point (x, y) and the point (4, 5), under the constraint that the point (x, y) lies on the line L. (The required distance is a minimum at the same point where its square is a minimum.) • In order to use Theorem 10.2.4 you need to write the function that you wish to minimize as a function of x alone. (Eliminate y by using the given constraint.) • In order to use the Method of Lagrange, write the function that you need to minimize as a function of x and y and also define an R 2 − R function g such that the given constraint is equivalent to the equation g (x, y) =