For each of the following functions, find the maximum and minimum values of the function on the circular disk: [math]. Do this by looking at the level curves and gradients.
(A) [math]:
maximum value =
minimum value =
(B) [math]:
maximum value =
minimum value =
(C) [math]:
maximum value =
minimum value =
Note: You can earn partial credit on this problem.
Find the maximum and minimum values of [math] on the ellipse [math].
maximum value =
minimum value =
Note: You can earn partial credit on this problem.
Find the maximum and minimum values of the function [math] subject to the constraint [math]. Maximum value is , occuring at points (positive integer or "infinitely many"). Minimum value is , occuring at points (positive integer or "infinitely many").
Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid [math] Hint: By symmetry, you can restrict your attention to the first octant (where [math]), and assume your volume has the form [math]. Then arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume:
The maximum value of [math] subject to the constraint [math] is [math]. The method of Lagrange multipliers gives [math]. Find an approximate value for the maximum of [math] subject to the constraint [math].}
[math]
What is the shortest distance from the surface [math] to the origin?
distance =
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STM32 (1)
