Question

Find two positive real numbers whose product is a maximum.
The sum of the first and three times the second is 30

Answer 1

x + 3y = 30
y = 10 - x/3
Maximize f(x) = x(10 - x/3) = 10x - (x^2)/3
f'(x) = (d/dx)(10x - (x^2)/3) = 10 - 2x/3
Maximum at f'(x) = 0
10 - 2x/3 = 0
x = 15
y = 10 - 15/3 = 5

Answer 2

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