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

Explanation

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}-70 y)(y^{5})`

`f(y)=1225 y^{5}+y^{7}-70 y^{6} `

`f^{prime}(y)=6125 y^{4}+7 y^{6}-420 y^{5}`

Putting this to zero we get

`y=35, y=25`

`f(y)=24500 y^{3}+42 y^{5}-2100 y^{4}`

...