The maximum value of the function $f(x)=3x^3-18x^2+27x-40$ on the set $S=\{x \in R : x^2+30 \leq 11x\}$ is

If $f(x) = x + \frac{1}{x}$,$x \neq 0$,then the local maximum and minimum values of the function $f$ are respectively....

If $PQ$ and $PR$ are the two sides of a triangle,then the angle between them which gives the maximum area of the triangle is

If the extreme value of the function $f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x}$ in the interval $[0, \frac{\pi}{2}]$ is $m$ and it exists at $x = k$,then $\cos k =$

The number of distinct real roots of the equation $3x^{4} + 4x^{3} - 12x^{2} + 4 = 0$ is ..... .

Let $f:[-1,1] \rightarrow R$ be defined as $f(x)=ax^{2}+bx+c$ for all $x \in[-1,1],$ where $a, b, c \in R$ such that $f(-1)=2, f^{\prime}(-1)=1$ and for $x \in(-1,1)$ the maximum value of $f^{\prime\prime}(x)$ is $\frac{1}{2}.$ If $f(x) \leq \alpha$ for all $x \in[-1,1],$ then the least value of $\alpha$ is equal to: