Answer to Question #20345 in Calculus for Sara

Question #20345
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.
1
Expert's answer
2012-12-11T09:24:42-0500
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 (-∞, -∞, +∞).

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