The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :

An open box is made from a cardboard measuring 12cm x 12cm by cutting off equal squares from the corner and turning up the side. Then maximum possible volume of the box would be:

The function $f:R\to R$ is defined by $f(x)=2x^3+5$, then which of the following statements are correct ?

(A) $f(x)$ has no local maximum value.

(B) $f(x)$ has no local minimum value.

(C) $f(x)$ has both local maximum and local minimum values.

(D) $f(x)$ has neither a local maximum value nor a local minimum value

Choose the correct answer from the options given below:

The interval in which the function $f(x) = x^2 - 3x + 7$ is increasing, is

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

A spherical ice ball is melting at the rate of 100 $\pi$ cm³/min. The rate at which its radius is decreasing when its radius is 15 cm, is

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: