This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the ex
treme values of the function subject to the given constraint. f(x1, x2, ..., xn) = x1 + x2 + ... + xn; x12 + x22 + ... + xn2 = 4
1 answer:


The Lagrangian is

with partial derivatives (all set equal to 0)

for
, and

Substituting each
into the second sum gives

Then we get two critical points,

or

At these points we get a value of
, i.e. a maximum value of
and a minimum value of
.
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