1. (a) Throughout Question 1 part (a), let f be the function given by f(x, y)...

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.

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milcah answered 1 year ago

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