The sum of the maximum and the minimum values of $f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 4$ in the interval $(0, 2)$ is:

For all real values of $x,$ the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,find the minimum value of $f(x)$.

Statement-$I$: The sequence $a_n = \frac{n^2}{n^3 + 200}, n \in N$ has its $7^{th}$ term as the largest term.
Statement-$II$: The function $f(x) = \frac{x^2}{x^3 + 200}$ attains a local maximum at $x = 7$.

The function $f(x) = x^2 + \frac{54}{x}$

The sum of the maximum and minimum values of the function $f(x) = |5x - 7| + [x^2 + 2x]$ in the interval $[\frac{5}{4}, 2]$,where $[t]$ denotes the greatest integer function $\leq t$,is: