Question

Consider the function f : R 2 → R defined by f(x, y) = 4 + x 3 + y 3 − 3xy.

(a)Compute the directional derivative of f at the point (a, b) = ( 1 2 , 1 2 ), in the direction u = ( √ 1 2 , − √ 1 2 ). At the point ( 1 2 , 1 2 ), is u the direction of steepest ascent, steepest descent, or neither? Justify your answer.

(b)Must f attain an absolute minimum and an absolute maximum on the rectangle D = [0, 2] × [0, 4]? Justify your answer.

(c)Calculate the rate of change of f along the curve r(t) = (t, t2 ), at t = −1.

(d) Classify the critical points of f using the second derivative test.

Answer 1