Maximize [tex]U=ab[/tex] subject to [tex]2a+b=48[/tex]
[tex]b = 48-2a[/tex]
[tex]U = ab = a(48-2a)[/tex]
[tex]U = -2(a^2 - 24 a) =-2(a^2 - 24 a + 144) + 2(144) = -2(a-12)^2+288[/tex]
That's clearly maximum when the squared term is zero, so
[tex]a=12[/tex]
[tex]b=48-2(12) = 24[/tex]
[tex]U=12(24)=288[/tex]
Answer: a=12, b=24, max 288
