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

The function $f$ is defined by $f(x) = x^p (1 - x)^q$ for all $x \in R$,where $p, q$ are positive integers. The function has a maximum value for $x$ equal to:

If the derivative of a function $f$ is given by $f'(x) = (x - a)^{2m} (x - b)^{2n + 1}$,where $m$ and $n$ are positive integers and $a > b$,then which of the following is true?

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

Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5$.

In the interval $[0, 2\pi]$,the slope of the tangent to the function $f(x) = e^x \sin x$ is maximum at $x = \dots$