Question

Two numbers (represented by A and B) when multiplied together produce a maximum value. Two times the first number (A) plus one times the second number (B) must alejas equal 48.

Answer 1

Maximize $U=ab$ subject to $2a+b=48$

$b = 48-2a$

$U = ab = a(48-2a)$

$U = -2(a^2 - 24 a) =-2(a^2 - 24 a + 144) + 2(144) = -2(a-12)^2+288$

That's clearly maximum when the squared term is zero, so

$a=12$

$b=48-2(12) = 24$

$U=12(24)=288$

Answer: a=12, b=24, max 288