Find the largest product the positive numbers x,y,z can have if (x^2)+y+(z^2) = 16 .

Answer 1

f(x, y, z) = xyz;

Lagrange function F(x, y, z) = xyz - λ(x² + y + z² - 16);

F_x = yz - 2λx = 0; F_y = xz - λ = 0; F_z = xy - 2λz = 0; F_λ = - x² - y - z² + 16 = 0.λ

λ = xz; yz - 2x²z = 0; xy - 2xz² = 0; z = 0 or y = 2x², x = 0 or y = 2z²; x² = z².

If x = z = 0, then y = 16 and f(x, y, z) = 0.

Now, - x² - 2x² - x² = - 16; x² = 4 x = ±2, z = ±2 and y = 8.

So, max(xyz) = 2·8·2 = 32