This extreme value problem has a solution with both a maximum value and a minimum value. use lagrange multipliers to find the extreme values of the function subject to the given constraint. f (x, y, z) = 10x + 10y + 4z; 5x2 + 5y2 + 4z2 = 44

The expression

10 x + 10 y + 4 z + λ (5 x^2 + 5 y^2 + 4 z^2 - 44)

has partial derivatives with respect to x, y, z, and λ of

10 + 10 x λ

10 + 10 y λ

4 + 8 z λ

-44 + 5 x^2 + 5 y^2 + 4 z^2

Setting these to zero and solving simultaneously gives the solutions

(x, y, z, λ) = (2, 2, 1, - 1/2)

(x, y, z, λ) = (-2, - 2, - 1, 1/2)

The corresponding extreme values of f (x, y, z) are 44 and - 44.