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

If $y = a \ln |x + 1| + b(x + 1)^2 + x$ has an extremum value of $4$ at $x = 0$,then the value of $(a, b)$ is:

The cost function at American Gadget is $C(x) = x^3 - 6x^2 + 15x$ (where $x$ is in thousands of units and $x > 0$). The production level at which the average cost is minimum 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

If the function $f(x) = \frac{t + 3x - x^2}{x - 4}$,where $t$ is a parameter,has a local maximum and a local minimum,then the range of values of $t$ is:

The function $f(x) = 2x^3 - 3x^2 - 12x + 4$ has