Use the chain rule to find the partial derivative მყ of the composite function g(x, y) = = x³ + y³, where x(s, t) = s+t, y (s, t) = st. Express your answers in terms of s and t. (You need not simplify the final expression.) Let the real-valued function of two variables f(x, y) be defined by f(x, y) = 2x²-2xy +5y²+18x+1. a) Show that the function has exactly one stationary point at (-5, -1). b) Calculate the values of the second order partial derivatives of f at (-5,-1). c) Classify the stationary point (-5, -1). Use the method of Lagrange multipliers to find the maximum value of the function g(x, y) = xy subject to the constraint that the point (x, y) lies on the ellipse 1½ x² + 2y² = 1. State all the points at which the maximum occurs. Consider the function h: R2 → R, defined by h(x, y) = log (x² + y²). a) Find the linear Taylor approximation of h in the neighbourhood of point a = (1,0). b) Find the quadratic Taylor approximation of h in the neighbourhood of point a = (1,0). Consider the region R in the xy-plane given by R = {(x, y) = R² : x ≥ 0, y ≥ 0 and 1 < x² + y² ≤ 4} . a) Sketch the region R.