If $f(x) = 3x + \frac{12}{x}$ is continuous on $R - \{0\}$ and $M$ is its local maximum value,then $\lim_{x \rightarrow M} f(x) = $

If $f(x)$ is a non-zero polynomial of degree four,having local extreme points at $x = -1, 0, 1$; then the set $S = \{x \in R; f(x) = f(0)\}$ contains exactly

The set of all real values of $\lambda$ for which the function $f(x) = (1 - \cos^2 x)(\lambda + \sin x)$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ has exactly one maxima and exactly one minima is

$A(1,15), B(3,-12), C(6,12)$ are three consecutive turning points of a continuous curve $y=f(x)$. If $f(x)=0$ only for $x=\alpha$ and $x=\beta$,then $|\beta-\alpha| < $

If $h(x) = f(x) + f(-x)$,then at the point where $h(x)$ attains an extremum,$f'(x) - f'(-x)$ is equal to:

Slope of the tangent to the curve $y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$ where $0 \leq x \leq 2 \pi$ is minimum at $x=$