If $f(x) = x^4 - \dfrac{2x^3}{3}$, then $f(x)$ is

A. increasing in $\left(\dfrac{1}{2}, \infty\right)$

B. decreasing in $\left(\dfrac{1}{2}, \infty\right)$

C. increasing in $(-\infty, 0)$

D. decreasing in $(-\infty, 0) \cup \left(0, \dfrac{1}{2}\right)$

Choose the correct answer from the options given below:

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true?

(A) $f$ is increasing in $(-\infty, -2)$

(B) $f$ is increasing in $(-2, 3)$

(C) $f$ is decreasing in $(-\infty, -2)$

(D) $f$ is decreasing in $(3, \infty)$

Choose the correct answer from the options given below:

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :

In a sphere the rate of change of volume with respect to(w.r.t) time is

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is: