Answer to Question #20345 in Calculus for Sara
The sum of three numbers is 81. One number is twice the other. Find all three numbers so that their product is as large as possible.
Let's denote these numbers by x, y and z. Then we obtain the following system:
x+y+z = 81
y = 2x
F(x,y,z) = x·y·z --> max
We can rewrite function F(x,y,z) in the next form:
F(x) = x · 2x · (81 - x - 2x) = 2x²(81-3x).
F(x) grows indefinitely while x indefinitely decreases. So, the set of numbers that their product is as large as possible is (-∞, -∞, +∞).