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

If $x + y = 8$,find the maximum value of $xy$.

The function $f(x) = x e^{-x}$ for all $x \in R$ attains a maximum value at $x = k$, then $k = $

Let $f(x) = x \sin \pi x$,$x > 0$. Then for all natural numbers $n$,$f^{\prime}(x)$ vanishes at
$(A)$ a unique point in the interval $\left(n, n+\frac{1}{2}\right)$
$(B)$ a unique point in the interval $\left(n+\frac{1}{2}, n+1\right)$
$(C)$ a unique point in the interval $(n, n+1)$
$(D)$ two points in the interval $(n, n+1)$

If $ab = 2a + 3b$ where $a > 0$ and $b > 0$,then the minimum value of $ab$ is:

The maximum area (in sq. units) of a rectangle having its base on the $x-$axis and its other two vertices on the parabola $y = 12 - x^2$ such that the rectangle lies inside the parabola,is