The function $f(x) = \sin x + \cos x$; $0 \leq x \leq 2\pi$ is

(A) Increasing in $\left(0, \frac{\pi}{4}\right)$

(B) Decreasing in $(0, \pi)$

(C) Increasing in $\left(\frac{5\pi}{4}, 2\pi\right)$

(D) Decreasing in $\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$

(E) Neither increasing Nor decreasing in $\left(0, \frac{\pi}{4}\right)$

Choose the correct answer from the options given below:

Which of the following are correct?

(A) The function $f(x) = 3x+12$ is increasing on R.

(B) The function $f(x) = e^{2x}$ is decreasing on R.

(C) The function $f(x) = x^2-x-1$ is neither increasing nor decreasing on (-1, 1).

(D) The function $f(x) = x^3-3x^2+4x$ is increasing on R.

Choose the correct answer from the options given below:

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

Let height and radius of a right circular cylinder are $h$ and $r$ respectively. If it is open at the top, having a given surface area and greatest volume, then

The interval in which the function given by $f(x) = x^2 e^{-x}$ is strictly increasing is:

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is $\tan^{-1}\left(\frac{2}{3}\right)$. Water is poured into it at a constant rate 6 cubic metre per hour. The rate at which the level of the water is rising at the instant when the depth of water in the tank is 5 meter is