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

A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are

If a manufacturer's total cost 'C' is given by $C = \dfrac{x^2}{25} + 3x$, then the marginal cost when 7 units are produced will be:-

The difference of absoulte maximum (largest) and absoulte minimum (smallest) values of the function $f(x) = 12x^{4/3} - 6x^{1/3}, x \in [-1, 1]$ is

For the function $f(x) = \dfrac{3}{2}x^4 - 4x^3 - 45x^2 + 49$. Which of the following statements is/are TRUE ?

A. $f(x)$ is increasing in $(-\infty, -3)$

B. $f(x)$ is increasing in $(-3, 0) \cup (5, \infty)$

C. $f(x)$ is decreasing in $(-\infty, -3) \cup (0, 5)$

D. $f(x)$ is decreasing in $(-\infty, -3) \cup (5, \infty)$

Choose the correct answer from the options given below:

Rate of change of area of a circle with respect to radius

(A) When r = 2 cm is $4\pi$ cm²/cm.

(B) When r = 6 cm is $12\pi$ cm²/cm.

(C) When r = 12 cm is $6\pi$ cm²/cm.

(D) When r = 16 cm is $8\pi$ cm²/cm.

Choose the correct answer from the options given below:

The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :