Question

The sum of two positive numbers is $20$. If the product of the square of one and the cube of the other is maximum then they are _____

• A. $8,12$
• B. $12,9$
• C. $4,16$
• D. $7,13$

Answer 1

The correct option is A $8,12$

Let the ture positive numbers be $x,(x,y>0)$

$x+y=20$ - Given

We need to maximise $x^2{y}^3$

$x=20-y$

$f\left(y\right)=y^3(20-y{)}^2$

for minimum

$f^{\prime }\left(y\right)=0$

$3y^2(20-y{)}^2+y^{3}2(20-y\left)\right(-1)=0$

$3y^2(20-y{)}^2-2y^3(20-y)=0$

$y^2(20-y)\left(3\right(20-y)-2y)=0$

$60-3y-2y=0$ $y\ne 0\&y\ne 20$

$y=12$ if $y=20$ then $x=0$ which does not

$x=8$ make the $x^2{y}^3$ maximum