If $f(x) = x^2e^{-2x}, x > 0$,then the maximum value of $f(x)$ is ......

Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then

Find the points at which the function $f$ given by $f(x)=(x-2)^{4}(x+1)^{3}$ has
$(i)$ local maxima
$(ii)$ local minima
$(iii)$ point of inflexion

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

Let $S=(-1, \infty)$ and $f: S \rightarrow R$ be defined as $f(x)=\int_{-1}^x (e^t-1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61} dt$. Let $p$ be the sum of the squares of the values of $x$ where $f(x)$ attains local maxima on $S$,and $q$ be the sum of the values of $x$ where $f(x)$ attains local minima on $S$. Then,the value of $p^2+2q$ is

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$