For all $x \in \mathbb{R}$, the minimum value $\frac{1}{3}$ and the maximum value $3$ of $f(x) = \frac{x^2+x+1}{x^2-x+1}$ occur at $l$ and $m$ respectively. Then $l+m$ is equal to:

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is equal to

The minimum value of the function $f(x) = x \log x$ is

The acceleration $f \text{ ft/sec}^2$ of a particle after a time $t \text{ sec}$ starting from rest is given by $f = 6 - \sqrt{1.2t}$. Then the maximum velocity $v$ and time $T$ to attain this velocity are:

Statement-$I$: $e^{\pi} > \pi^e$.
Statement-$II$: The function $f(x) = x^{1/x}$ attains its global maximum at $x = e$.

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter $6$ units is