Two positive numbers x and y whose sum is 25 and the product $x^3y^2 $ is maximum are :

$x=15, y = 10$

The correct answer is Option (2) → $x=15, y = 10$

$x+y=25⇒y=25-x$

and,

$x^3y^2=x^3(25-x)^2$

$=x^3(625+x^2-50x)$

$=625x^3+x^5-50x^4$

$\frac{d(625x^3+x^5-50x^4)}{dx}=1875x^2+5x^4-200x^3$

for critical point,

$⇒1875x^2+5x^4-200x^3=0$

$⇒x^4-40x^3+375x^2=0$

$⇒x^2(25-x)(75-5x)=0$

Since $x>0$ and $y>0$,

$75-5x=0$

$5x=75⇒x=15$ and $y=10$