Find the extreme values of the function subject to the given constraint.
-$$f ( x , y , z ) = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$$
A) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: 0 at (0, 0, 0)
B) Maximum: 8 at (2, 0, 0); minimum: 0 at (0, 0, 0)
C) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2)
D) Maximum: 8 at (2, 0, 0); minimum: -8 at (-2, 0, 0)

Question

Quizplus · Accepted Answer

Using Lagrange multiplier method, we have:
$$3x^{2} = 2\lambda x$$
$$3y^{2} = 2\lambda y$$
$$3z^{2} = 2\lambda z$$
$$x^{2} + y^{2} + z^{2} = 4$$
By solving above equations, we get 4 critical points: (2,0,0), (0,2,0), (0,0,2), (-2,0,0), (0,-2,0), and (0,0,-2). We can then plug in these points into the function $f(x,y,z)$ to see which points give us the maximum and minimum values. It turns out that the maximum value is 8 which occurs at points (2,0,0), (0,2,0), and (0,0,2). The minimum value is -8 which occurs at points (-2,0,0), (0,-2,0), and (0,0,-2). Therefore, the best choice is C.

Find the Extreme Values of the Function Subject to the Given