CBSE - Class 12 Mathematics Application of Derivatives Worksheet
1.
Prove that the function $f$ given by $f(x) = x^2 – x + 1$ is neither strictly increasing nor decreasing on $(– 1, 1)$.
2.
An edge of a variable cube is increasing at the rate of $3$ cm/s. How fast is the volume of the cube increasing when the edge is $10$ cm long?
3.
At what points in the interval $[0, 2\pi]$, does the function $\sin 2x$ attain its maximum value?
4.
The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 – 0.003x^2 + 15x + 4000$. Find the marginal cost when $17$ units are produced.
5.
The radius of a circle is increasing uniformly at the rate of $3$ cm/s. Find the rate at which the area of the circle is increasing when the radius is $10$ cm.
6.
Prove that the following functions do not have maxima or minima: (ii) $g(x) = \log x$
7.
Prove that the logarithmic function is increasing on $(0, \infty)$.
8.
Find the maximum and minimum values, if any, of the following functions given by (iii) $f(x) = – (x – 1)^2 + 10$
9.
Find both the maximum value and the minimum value of $3x^4 – 8x^3 + 12x^2 – 48x + 25$ on the interval $[0, 3]$.
10.
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10$ cm.
11.
Show that the function given by $f(x) = \sin x$ is (a) increasing in $(0, \frac{\pi}{2})$
12.
Find the intervals in which the following functions are strictly increasing or decreasing: (d) $6 – 9x – x^2$
13.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $\tan^{-1} \sqrt{2}$.
14.
Prove that the following functions do not have maxima or minima: (iii) $h(x) = x^3 + x^2 + x +1$
15.
Find the maximum and minimum values, if any, of the following functions given by (iv) $g(x) = x^3 + 1$
16.
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: (iv) $f(x) = \sin x – \cos x, 0 < x < 2\pi$
17.
Find the maximum profit that a company can make, if the profit function is given by $p(x) = 41 – 72x – 18x^2$
18.
A rectangular sheet of tin $45$ cm by $24$ cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?
19.
A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15$ cm.
20.
Prove that the function $f$ given by $f(x) = \log \sin x$ is increasing on $(0, \frac{\pi}{2})$ and decreasing on $(\frac{\pi}{2}, \pi)$.
