x and y are positive values satisfying x + y = 1/4. When ln(y^2+4xy+7) attains its maximum (subject to the given constraint), what is the value of 1/xy?

Answer 1

well, since x+y = 1/4, y=1/4 - x
ln(y^2 + 4xy + 7)
= ln((1/4 - x)^2 + 4x(1/4 -x) + 7)
= ln((113+8x-48x^2)/16)

sinc lna > lnb if a>b, we want the maximum value for 113+8x-48x^2

That occurs when x = 1/12, y=1/6
So, 1/xy = 72