1. (a) Throughout Question 1 part (a), let f be the functiongiven by f(x, y) = 6+x^3+y^3?3xy.
(i) At the point (0, 1), in what direction does the function fhave the largest directional derivative?
(ii) Find the directional derivative of the function f at thepoint (0, 1) in the direction of the vector [3, 4] .
(iii) The function f has critical points at (0, 0) and at (1,1). Classify the natures of these critical points by using theHessian. Justify your answer.
(iv) Suppose that x = t^2 and y = 1 ? t^3 . Use the chain ruleto calculate df/dt. You should write your function as a function oft but there is no need to simplify your answer.
(b) Consider optimisation of the function f(x, y) = 4x ? 2ysubject to the constraint x^2 + y^2 = 125. Use the method ofLagrange multipliers to find the critical points of thisconstrained optimisation problem. You do not need to determine thenature of the critical points.
