Let $f(x) = 2 \cos^{-1} x + 4 \cot^{-1} x - 3x^2 - 2x + 10$,where $x \in [-1, 1]$. If $[a, b]$ is the range of the function,then $4a - b$ is equal to:

For the function $f(x) = x^4(12\ln x - 7)$,match the following columns:

Column-$I$	Column-$II$
$(A)$ If $(a, b)$ is a point of inflection,then $a - b$ equals	$(P)$ $3$
$(B)$ If $e^t$ is the point of local minimum,then $12t$ equals	$(Q)$ $1$
$(C)$ If the graph is concave down on $(d, e)$,then $d + 3e$ equals	$(R)$ $4$
$(D)$ If the graph is concave up on $(p, \infty)$,then $p$ equals	$(S)$ $8$

The function $f(x) = x^2 + \frac{54}{x}$

The lengths of the sides of a triangle are $10+x^2$,$10+x^2$ and $20-2x^2$. If for $x=k$,the area of the triangle is maximum,then $3k^2$ is equal to

Let $f(x)$ be a polynomial of degree $4$ having extreme values at $x = 1$ and $x = 2$. If $\mathop {\lim }\limits_{x \to 0} \left( {\frac{{f(x)}}{{{x^2}}} + 1} \right) = 3$,then $f(-1)$ is equal to

The local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is