CBSE Class 12 Mathematics — Application of Derivatives

The least value of $f(x) = x^3 - 12x,\ x \in [0, 3]$ is

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of $1 \text{ mm}^3/\text{min}$. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is $10 \text{ mm}$, if the semi-vertical angle of conical bottle is $\pi/6$.

How many subscribers will discontinue after an increase of Rs x in annual fee?

If $R(x)$ denotes the total revenue collected after the increase of Rs x in subscription fee, express $R(x)$ as a function of x.

(a) Find the value of x for which $R(x)$ is maximum. OR (b) Find the sub-intervals of $(0, 5000)$ in which $R(x)$ is increasing and decreasing.