CBSE - Class 12 Mathematics Application of Derivatives Worksheet
1.
At what points in the interval $[0, 2\pi]$, does the function $\sin 2x$ attain its maximum value?
2.
Prove that the following functions do not have maxima or minima: (iii) $h(x) = x^3 + x^2 + x +1$
3.
A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2x + 1)$. Find the rate of change of its volume with respect to $x$.
4.
Let $I$ be any interval disjoint from $[–1, 1]$. Prove that the function $f$ given by $f(x) = x + \frac{1}{x}$ is increasing on $I$.
5.
Find the maximum and minimum values, if any, of the following functions given by (iv) $f(x) = | \sin 4x + 3|$
6.
Show that the function given by $f(x) = \sin x$ is (c) neither increasing nor decreasing in $(0, \pi)$
7.
Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^2 y^5$ is a maximum.
8.
For what values of $a$ the function $f$ given by $f(x) = x^2 + ax + 1$ is increasing on $[1, 2]$?
9.
Find the intervals in which the function $f$ given by $f(x) = 2x^2 – 3x$ is (a) increasing
10.
The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is
a.
$(2\sqrt{2},4)$
b.
$(2\sqrt{2},0)$
c.
$(0, 0)$
d.
$(2, 2)$
11.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iv) $f(x) = (x-1)^2 + 3, x \in [-3, 1]$
12.
The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width $y$ is increasing at the rate of $4$ cm/minute. When $x = 8$cm and $y = 6$cm, find the rates of change of (b) the area of the rectangle.
13.
Find the maximum value of $2x^3 – 24x + 107$ in the interval $[1, 3]$. Find the maximum value of the same function in $[–3, –1]$.
14.
Find the maximum and minimum values, if any, of the following functions given by (ii) $f(x) = 9x^2 + 12x + 2$
15.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (v) $f(x) = x^3 – 6x^2 + 9x + 15$
16.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iii) $f(x) = 4x - \frac{1}{2}x^2, x \in [-2, \frac{9}{2}]$
17.
The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13x^2 + 26x + 15$. Find the marginal revenue when $x = 7$.
18.
Find the rate of change of the area of a circle with respect to its radius $r$ when (b) $r = 4$ cm
19.
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
20.
Find the intervals in which the following functions are strictly increasing or decreasing: (d) $6 – 9x – x^2$
