A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is

(A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$

(B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$

(C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$

(D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$

(E) strictly increasing on $(0, \pi)$

Choose the correct answer from the options given below:

The equation of the tangent, to the curve $y = x^2 - 2x - 3$ which is perpendicular to the line $x + 2y + 3 = 0$, is

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is

Based on above information answer the following question :

$x$ and $y$ will satisfy :

The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

For the function $f(x) = e^x + e^{-x}$

(A) $f'(x) = e^x - e^{-x}$

(B) The critical point is $x = 0$

(C) The minimum value is 1

(D) $x = 0$ is the point of local minimum.

Choose the correct answer from the options given below: