If $f(x)=a \log |x|+b x^2+x$ has extreme values at $x=-1$ and $x=2$,then the ordered pair $(a, b)=$

Consider a square of side $2 \ cm$. It is cut from one of its corners as shown in the adjacent figure. The maximum value of the sum of the perimeters of the two plane figures thus formed is

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

Let $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}$. Then,which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g(x)$ is $2^{7/6}$
$(B)$ The maximum value of $g(x)$ is $1 + 2^{1/3}$
$(C)$ The function $g(x)$ attains its maximum at more than one point
$(D)$ The function $g(x)$ attains its minimum at more than one point

For the curve $y = x e^x$,the point:

$A$ stone moving vertically upwards has its equation of motion $s = 490t - 4.9t^2$. The maximum height reached by the stone is