hi, can u please provide full solutions for these questions?

+ 127% MATH2211 Multivariable Calculus Test 2 10 November 2020 Please write down your name and university number. Precise and adequate explanations should be given to each problem. 1. Let f, g : R - R be two smooth functions, and define h : R2 - R by h(x, y) = f(x)g(y). (a) (5 points) Find the second-order Taylor polynomial of h at the point (0,0) in terms of the functions f, g and their derivatives. (b) (3 points) Let P2(x) and Q2(y) be the second-order Taylor's polynomials of f(x) and g(y) at the point 0 respectively. Show that the terms in the polynomial P2(x) Q2(y) with total degree at most 2 is the same as the polynomial found in part (a). 2. (8 points) Define f : R2 - R by hero .co' f (x, y) = erty (2 2 + y ) Find all critical point(s) of f, and determine whether each critical point is a point of local minimum, a point of local maximum or a saddle point. (No need to check the global extrema.) 3. (8 points) Let k be a positive real number. Using the method of Lagrange multiplier, find the global maximum and global minimum of x - y, where a and y are real numbers satisfying x 2 + ky ? = 1