The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. The rate at which the radius of the balloon increases when the radius is 15 cm, is:

The function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$

(A) is increasing on $\left[0, \frac{\pi}{6}\right]$

(B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$

(C) is increasing on $\left[0, \frac{\pi}{2}\right]$

(D) is decreasing on $\left[0, \frac{\pi}{2}\right]$

Choose the correct answer from the options given below:

The function $f(x) = x^3 + 2x^2 - 1$ is

(A) increasing in $\left(\dfrac{-4}{3}, 0\right)$

(B) decreasing in $\left(\dfrac{-4}{3}, 0\right)$

(C) increasing in $\left(-\infty, \dfrac{-4}{3}\right)$

(D) increasing in $(0, \infty)$

Choose the correct answer from the options given below:

Let $f(x) = (x-a)^2+(x-b)^2+(x-c)^2$ then $f(x)$ has a minimum value at $x =$

Which of the following statements are true?

(A) The function $f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x$ has 3 critical points.

(B) The function $f(x) = |x| + 3$ has no minimum value.

(C) A local maximum value is always the absolute maximum value.

(D) $f(x) = x^2$ has minima at $x=0$.

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 :