Question

Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^2{y}^5$ is a maximum.

Answer 1

We have to maxims $x^2{y}^5$ and we have $x+y=35$

So put $x=35-y$ in $x^2{y}^5$

we get $(1225+y^2-70y)\left(y^5\right)$

$f\left(y\right)=1225y^5+y^7-70y^6$

$f^{{}^{\prime }}\left(y\right)=6125y^4+7y^6-420y^5$

Putting this to zero we get

$y=35,y=25$

$f^{{}^{\prime \prime }}\left(y\right)=24500y^3+42y^5-2100y^4$

$f^{{}^{\prime \prime }}\left(35\right)=24500\times {35}^3+42\times {35}^5-2100\times {35}^4$

$f^{{}^{\prime \prime }}\left(35\right)>0$

At $y=25$

$f^{{}^{\prime \prime }}\left(25\right)=24500\times {25}^3+42\times {25}^5-2100\times {25}^4$

$f^{{}^{\prime \prime }}\left(25\right)<0$

So given function will have maxima when $y=25$

So $x=10$