The function $f(x) = \sin x(1 + \cos x)$ at $x = \frac{\pi}{3}$ is:

$A$ cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is:

If the function $f$ given by $f(x) = x^3 - 3(a - 2)x^2 + 3ax + 7$ for some $a \in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$,then a root of the equation $\frac{f(x) - 14}{(x - 1)^2} = 0$ $(x \neq 1)$ is

$A$ wire of length $20 \text{ cm}$ is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is (in $\text{ cm}^2$)

If $p(x)$ is a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$,then $p(0)$ is equal to:

Find the local maximum and local minimum values of $f(x) = \sin 2x - x$ on the interval $[0, \pi]$.