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

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

If the sum of the lengths of hypotenuse and a side of a right angled triangle is given, then the area of the triangle is maximum when angle between them is:

The function $f(x) = 2x^3-15x^2+36x+4$ has local maxima at $x=$

The function $f(x) = -x^3 + 12x^2 - 36x + 21$ is

A. Increasing on $(-\infty, 2)$

B. Decreasing on $(-\infty, 2)$

C. Increasing on $(2, 6)$

D. Decreasing on $(6, \infty)$

Choose the correct answer from the options given below:

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then

(A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$

(B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$

(C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$

(D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$

Choose the correct answer from the options given below: