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

The function $f(x) = x^5 - 5x^4 + 5x^3 - 1$ is:

For the function $f(x) = \int_{0}^{x} \frac{\sin t}{t} dt$,where $x > 0$,which of the following is true?

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 $(n, n+1)$

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2 R}{\sqrt{3}} .$ Also find the maximum volume.