Find the largest product the positive numbers x, y, and z can have if 2x + y + z^2 = 20.

Question

Find the largest product the positive numbers x, y, and z can have if 2x + y + z² = 20.

Answer 1

Use LaGrange multipliers

xyz - λ(2x + y + z^2)

If you hate dealing with products, you may change this to maximizing

ln(x) + ln(y) + ln(z) - λ(2x + y + z^2)

We could put in the greater than or equal to 0 constraints, but we know that, for x, y, z positive, there will be an internal solution (one not applying the 0 constraints), as these constraints give a value to xyz of 0.

Using the modified problem, we have

1/x - 2λ = 0

1/y - λ = 0

1/z - 2λz = 0

Then√, we get that y = 1/λ , x = 1/(2λ), and z = √(1/2λ)

Then, as 2x + y + z^2 = 20, 2 * 1/(2λ) + 1/λ + 1/(2λ) = 20

5/(2λ) = 20

40λ = 5

λ = 1/8

y = 1/(1/8) = 8

x = 1/(2λ) = 1/(1/4) = 4

z = √(1/(2λ)) = √4 = 2

(x,y,z) = (4, 8, 2)

xyz = 64