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Date

Vector Calculus Independent Study

Unit 4 Sample Test

1. [25 points] Find the absolute max and min of f(x, y, z) = x − y + 3z on the ball x^{2} + y^{2} + z^{2} ≤ 4.

2. [25 points] Find and classify (as local min, local max, saddle, or undetermined) the critical points of f(x, y) = 3x + 12y − x^{3} − y^{3}.

3. [25 points] Find the point on the plane 4x − 3y + z = 5 closest to the point (2, 1, −1).

4. [25 points] Show that a rectangular box of a given volume has minimum surface area when the box is a cube.