Two positive numbers x and y such that x+y = 60 and xy^3 is maximum a…

Given

$x+y=60$

Let

$f(x)=xy^{3}$

$y=60-x$

$f(x)=x(60-x)^{3}$

$f'(x)=(60-x)^{3}+x\cdot3(60-x)^{2}(-1)$

$=(60-x)^{2}\left[(60-x)-3x\right]$

$=(60-x)^{2}(60-4x)$

$f'(x)=0$

$60-4x=0$

$x=15$

$y=60-15=45$

The two numbers are $x=15$ and $y=45$.