Find the positive value of $a$ for which the equality $2 \alpha + \beta = 8$ holds,where $\alpha$ and $\beta$ are the points of maximum and minimum,respectively,of the function $f(x) = 2 x^3 - 9 a x^2 + 12 a^2 x + 1$.

Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

$A$ circular sector of perimeter $60 \ m$ with maximum area is to be constructed. The radius of the circular arc in meters must be: (in $m$)

The coordinates of the point $P$ on the graph of the function $y = e^{-|x|}$ where the portion of the tangent intercepted between the coordinate axes has the greatest area,is

For all $x \in (0, 1)$,which of the following inequalities is true?