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