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.