If $x$ and $y$ are two positive integers such that $x + 2y = 10$ and $x^2 y^3$ is maximum,then $x^2 + 2y^3 =$

Local maximum value of the function $\frac{\log x}{x}$ is

If $a, b > 0$,then the minimum value of $y = \frac{b^2}{a-x} + \frac{a^2}{x}$ for $0 < x < a$ is

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\int_{0}^{x}e^{(x-t)}f(t)dt, x\in R$ and let $g(x)=\int_{0}^{x}(f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x\in R.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to . . . . . . .

$A$ rectangle has one side on the positive $y-$ axis and one side on the positive $x-$ axis. The upper right hand vertex lies on the curve $y = \frac{\ln x}{x^2}$. The maximum area of the rectangle is

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ unit}$ is (in square unit)