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1. (a) Throughout Question 1 part (a), let f be the function given by f(x, y) = 6+x^3+y^3−3xy.

(i) At the point (0, 1), in what direction does the function f have the largest directional derivative?

(ii) Find the directional derivative of the function f at the point (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 the Hessian. Justify your answer.

(iv) Suppose that x = t^2 and y = 1 − t^3 . Use the chain rule to calculate df/dt. You should write your function as a function of t but there is no need to simplify your answer.

(b) Consider optimisation of the function f(x, y) = 4x − 2y subject to the constraint x^2 + y^2 = 125. Use the method of Lagrange multipliers to find the critical points of this constrained optimisation problem. You do not need to determine the nature of the critical points.

Expert Solution

milcah answered 2 years ago

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