Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) f(x, y, z) = x2 + y2 + z2; x4 + y4 + z4 = 13

The Lagrangian

has critical points where the first derivatives vanish:

We can't have , since that contradicts the last condition.

(0 critical points)

If two of them are zero, then the remaining variable has two possible values of . For example, if , then .

(6 critical points; 2 for each non-zero variable)

If only one of them is zero, then the squares of the remaining variables are equal and we would find (taking the negative root because must be non-negative), and we can immediately find the critical points from there. For example, if , then . If both are non-zero, then , and

and for either choice of , we can independently choose from .

(12 critical points; 3 ways of picking one variable to be zero, and 4 choices of sign for the remaining two variables)

If none of the variables are zero, then . We have

and similary have the same solutions whose signs can be picked independently of one another.

(8 critical points)

Now evaluate at each critical point; you should end up with a maximum value of and a minimum value of (both occurring at various critical points).

Here's a comprehensive list of all the critical points we found:

answer: 12

step-by-step explanation: