At what value of $x$ does the function $f(x) = \sin x(1 + \cos x)$ have a maximum value?

Find all points of local maxima and local minima of the function $f$ given by $f(x) = x^3 - 3x + 3$.

The number of points in $(-\infty, \infty)$,for which $x^2-x \sin x-\cos x=0$,is

Let $a, b \in R$ be such that the function $f(x) = \ln|x| + bx^2 + ax, x \neq 0$ has extreme values at $x = -1$ and $x = 2$.
Statement-$1$: $f$ has a local maximum at $x = -1$ and $x = 2$.
Statement-$2$: $a = \frac{1}{2}$ and $b = -\frac{1}{4}$.

Let $f(x) = \alpha x^2 - 2 + \frac{1}{x}$,where $\alpha$ is a real constant. The smallest $\alpha$ for which $f(x) \geq 0$ for all $x > 0$ 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| < $