The sum of the maximum and minimum values of the function $f(x)=\frac{x^2-x+1}{x^2+x+1}$ is

If the sum of two numbers is $3$,then the maximum value of the product of the first and the square of the second is:

If $0 < x < \frac{\pi}{2}$,then the maximum area (in sq. units) of the triangle whose vertices are $(0,0)$,$(x, \cos x)$ and $(\sin^3 x, 0)$ is

The minimum value of $e^{(2x^2 - 2x - 1)\sin^2 x}$ is .......

Let $S=(-1, \infty)$ and $f: S \rightarrow R$ be defined as $f(x)=\int_{-1}^x (e^t-1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61} dt$. Let $p$ be the sum of the squares of the values of $x$ where $f(x)$ attains local maxima on $S$,and $q$ be the sum of the values of $x$ where $f(x)$ attains local minima on $S$. Then,the value of $p^2+2q$ is

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