If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function $f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x \in R$,then

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

If $f(x) = 1 + 2x^2 + 2^2 x^4 + \dots + 2^{10} x^{20}$,then $f(x)$ has:

If the function $f(x) = a \sin(x) + \frac{1}{3} \sin(3x)$ attains its maximum value at $x = \frac{\pi}{3}$,then $a$ equals:

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$,then the ordered pair $(m, M)$ is equal to

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,