The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is .....

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:

If $x=1$ and $x=2$ are extreme points of $f(x)=\alpha \log x+\beta x^2+x$,where $\alpha$ and $\beta$ are constants,then the value of $\alpha^2+2 \beta$ is

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

The vertices of a triangle are $(0, 0)$,$(x, \cos x)$,and $(\sin^3 x, 0)$ where $0 < x < \frac{\pi}{2}$. The maximum area for such a triangle in sq. units is:

The sum of absolute maximum and absolute minimum values of the function $f(x)=|2 x^{2}+3 x-2|+\sin x \cos x$ in the interval $[0,1]$ is