Application of Derivative
Showing all questions for Class 12 Math
Question 1:
Find the rate of change of the area of a circle with respect to its radius \( r \) when (a) \( r = 3 \) cm (b) \( r = 4 \) cm...
Question 2:
The volume of a cube is increasing at ( 8 ) cm³/s. How fast is the surface area increasing when the edge length is ( 12 ) cm?...
Question 3:
The radius of a circle is increasing uniformly at the rate of \(3\;{\text{cm}}/{\text{s}}\). Find the rate at which the area of the circle is increasing when the radius is \(10\;{\text{cm}}/{\text{s}}\)...
Question 4:
An edge of a variable cube is increasing at the rate of \(3\;{\text{cm}}/{\text{s}}\). How fast is the volume of the cube increasing when the edge is \(10\;{\text{cm}}\) long?...
Question 5:
A stone is dropped into a quiet lake and waves move in circles at the speed of \(5\;{\text{cm}}/{\text{s}}.\) At the instant when the radius of the circular wave is \(8\;{\text{cm}}\), how fast is the enclosed area increasing?...
Question 6:
The radius of a circle is increasing at the rate of \(0.7,\text{cm/s}\). What is the rate of increase of its circumference?...
Question 7:
The length \(x\) of a rectangle is decreasing at the rate of \(5,\text{cm/min}\), and the width \(y\) is increasing at the rate of \(4,\text{cm/min}\). When \(x = 8,\text{cm}\) and \(y = 6,\text{cm}\), find the rate of change of:(a) Perimeter (b) Are...
Question 8:
A spherical balloon is being inflated at \(900,\text{cm}^3/\text{s}\). Find the rate of change of radius when \(r = 15,\text{cm}\).
...
Question 9:
A spherical balloon has a variable radius. Find the rate at which its volume is increasing with respect to radius when \(r = 10,\text{cm}\).
...
Question 10:
A ladder \(5,\text{m}\) long leans against a wall. The bottom slides away at \(2,\text{cm/s}\). How fast is the top of the ladder sliding down when the bottom is \(4,\text{m}\) from the wall?
...
Question 11:
A particle is moving along the curve \(6y = x^3 + 2\).
Find the points on the curve where the \(y\)-coordinate is changing 8 times as fast as the \(x\)-coordinate....
Question 12:
The radius of an air bubble is increasing at the rate of \(\dfrac{1}{2}\;{\text{cm}}/{\text{s}}\). At what rate is the volume of the bubble increasing when the radius is \(1\;{\text{cm}}\) ? ...
Question 13:
A balloon, which always remains spherical, has a variable diameter \(\dfrac{3}{2}(2x + 1)\). Find the rate of change of its volume with respect to \(x\)....
Question 14:
Sand is pouring from a pipe at the rate of \(12\;{\text{c}}{{\text{m}}^3}/{\text{s}}\). The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the...
Question 15:
The total cost \(C(x)\) in Rupees associated with the production of \(x\) units of an item is given by \(C(x) = 0.007{x^3} - 0.003{x^2} + 15x + 4000\). Find the marginal cost when 17 units are produced....
Question 16:
The total revenue in Rupees received from the sale of \(x\) units of a product is given by \(R(x) = 13{x^2} + 26x + 15\). Find the marginal revenue when \(x = 7\)....
Question 17:
The rate of change of the area of a circle with respect to its radius \(r\) at \(r = 6\;{\text{cm}}\) is...
Question 18:
The total revenue in Rupees received from the sale of \(x\) units of a product is given by \(R(x) = 3{x^2} + 36x + 5\). The marginal revenue, when \(x = 15\) is ...
Question 19:
Show that the function given by \(f(x) = 3x + 17\) is strictly increasing on \(\mathbb{R}\)....
Question 20:
Show that the function \(f(x) = e^{2x}\) is strictly increasing on \(\mathbb{R}\)....
Question 21:
Show that the function given by \(f(x) = \sin x\) is \[\](a) Strictly increasing in \(\left( {0,\dfrac{\pi }{2}} \right)\) \[\](b) Strictly decreasing \(\left( {\dfrac{\pi }{2},\pi } \right)\) \[\] (c) Neither increasing nor decreasing in \((0,\pi )\...
Question 22:
Find the intervals in which the function \(f(x) = 2x^2 - 3x\) is:
(a) strictly increasing
(b) strictly decreasing
...
Question 23:
Find the intervals in which the function
\(f(x) = 2x^3 - 3x^2 - 36x + 7\)
is:
(a) Strictly increasing
(b) Strictly decreasing...
Question 24:
Find the intervals in which the function
\(f(x) = x^2 + 2x - 5\)
is strictly increasing or strictly decreasing....
Question 25:
Find the intervals in which the function
\(f(x) = 10 - 6x - 2x^2\)
is strictly increasing or strictly decreasing....
Question 26:
Find the intervals in which the function
\(f(x) = -2x^3 - 9x^2 - 12x + 1\)
is strictly increasing or strictly decreasing....
Question 27:
Find the intervals in which the function
\(f(x) = 6 - 9x - x^2\)
is strictly increasing or strictly decreasing....
Question 28:
Find the intervals in which the function
\(f(x) = (x + 1)^3(x - 3)^3\)
is strictly increasing or strictly decreasing.
...
Question 29:
Show that \( y = \log (1 + x) - \dfrac{2x}{2 + x},\ x > -1 \), is an increasing function throughout its domain....
Question 30:
Find the values of \(x\) for which \( y = [x(x - 2)]^2 \) is an increasing function....
Question 31:
Prove that \( y = \dfrac{4\sin \theta}{2 + \cos \theta} - \theta \) is an increasing function of \( \theta \) in \( \left[0, \dfrac{\pi}{2} \right] \).
...
Question 32:
Prove that the logarithmic function is strictly increasing on \( (0, \infty) \)....
Question 33:
Prove that the function \( f \) given by \( f(x) = x^2 - x + 1 \) is neither strictly increasing nor strictly decreasing on \( (-1, 1) \).
...
Question 34:
Find the least value of \( a \) such that the function
\(
f(x) = x^2 + ax + 1
\)
is strictly increasing on the interval \( (1, 2) \)....
Question 35:
Let \( \mathbf{I} \) be any interval disjoint from \( (-1,1) \). Prove that the function \( f \) given by
\(
f(x) = x + \frac{1}{x}
\)
is strictly increasing on \( \mathbf{I} \)....
Question 36:
Prove that the function \( f \) given by
\(
f(x) = \log \sin x
\)
is strictly increasing on \( \left(0, \dfrac{\pi}{2} \right) \) and strictly decreasing on \( \left( \dfrac{\pi}{2}, \pi \right) \).
...
Question 37:
Prove that the function \( f \) given by
\(
f(x) = \log \cos x
\)
is strictly decreasing on \( \left(0, \dfrac{\pi}{2} \right) \) and strictly increasing on \( \left( \dfrac{\pi}{2}, \pi \right) \).
...
Question 38:
Prove that the function given by
\(
f(x) = x^3 - 3x^2 + 3x = 100
\)
is increasing on \( \mathbb{R} \).
...
Question 39:
Find the maximum and minimum values, if any, of the function
\(
f(x) = (2x - 1)^2 + 3
\)...
Question 40:
Find the maximum and minimum values, if any, of the function
\(
f(x) = 9x^2 + 12x + 2
\)...
Question 41:
Find the maximum and minimum values, if any, of the function
\(
f(x) = - (x - 1)^2 + 10
\)...
Question 42:
Find the maximum and minimum values, if any, of the function
\(
g(x) = x^3 + 1
\)...
Question 43:
Find the maximum and minimum values of the function
\(
f(x) = |x + 2| - 1
\)...
Question 44:
Find the maximum and minimum values of the function
\(
g(x) = -|x + 1| + 3
\)...
Question 45:
Find the maximum and minimum values of the function
\(
h(x) = \sin (2x) + 5
\)...
Question 46:
Find the maximum and minimum values of the function
\(
f(x) = |\sin(4x) + 3|
\)
...
Question 47:
Find the maximum and minimum values of the function
\(
h(x) = x + 4, \quad x \in (-1, 1)
\)...
Question 48:
Find the local maxima and local minima, if any, of the function \( f(x) = x^2 \). Find also the local maximum and the local minimum values.
...
Question 49:
Find the local maxima and local minima, if any, of the function \( g(x) = x^3 - 3x \).
...
Question 50:
Find the local maxima and minima of \( h(x) = \sin x + \cos x \) in \( \left( 0, \frac{\pi}{2} \right) \)
...
Question 51:
Find the local maxima and minima of \( f(x) = \sin x - \cos x \) in \( \left(0, 2\pi \right) \).
...
Question 52:
Find the local maxima and minima of \( f(x) = x^3 - 6x^2 + 9x + 15 \).
...
Question 53:
Find the local maxima and minima of \(g(x) = \frac{x}{2} + \frac{2}{x}, \, x > 0 \).
...
Question 54:
Find the local maxima and minima of \( g(x) = \frac{1}{x^2 + 2} \).
...
Question 55:
Find the local maxima and minima of \( f(x) = x \sqrt{1 - x}, \, x > 0 \).
...
Question 56:
Prove that the function \( f(x) = e^x \) does not have any maximum or minimum.
...
Question 57:
Prove that the function \( g(x) = \log x \) does not have any maximum or minimum.
...
Question 58:
Prove that the function \( h(x) = x^3 + x^2 + x + 1 \) does not have any maximum or minimum.
...
Question 59:
Find the absolute maximum and minimum values of the function \( f(x) = x^3 \) in the interval \([ -2, 2 ]\).
...
Question 60:
Find the absolute maximum and minimum values of the function \( f(x) = \sin x + \cos x \) in the interval \( [0, \pi] \).
...
Question 61:
Find the absolute maximum and minimum values of the function \( f(x) = 4x - \frac{1}{2}x^2 \) in the interval \( \left[ -2, \frac{9}{2} \right] \).
...
Question 62:
Find the absolute maximum and minimum values of the function \( f(x) = (x - 1)^2 + 3 \) in the interval \( [-3, 1] \).
...
Question 63:
Find the maximum profit that a company can make, if the profit function is given by
\( p(x) = 41 - 24x - 18x^2 \)
...
Question 64:
Find the intervals in which the function
\( f(x) = x^3 + \frac{1}{x^3}, \quad x \ne 0 \)
is (i) Increasing (ii) Decreasing.
...
Question 65:
At what points in the interval \([0, 2\pi]\), does the function
\( f(x) = \sin 2x \)
attain its maximum value?
...
Question 66:
What is the maximum value of the function \(\sin x + \cos x\) ?...
Question 67:
Find the maximum value of \(2x^3 - 24x + 107\) in the interval \([1,3]\).
Find the maximum value of the same function in \([-3, -1]\)....
Question 68:
It is given that at \(x = 1\), the function \(f(x) = x^4 - 62x^2 + ax + 9\) attains its maximum value on the interval \([0, 2]\).
Find the value of \(a\).
...
Question 69:
Find the maximum and minimum values of \( f(x) = x + \sin 2x \) on the interval \([0, 2\pi]\).
...
Question 70:
Find two numbers whose sum is 24 and whose product is as large as possible.
...
Question 71:
Find two positive numbers \( x \) and \( y \) such that \( x + y = 60 \) and \( x y^3 \) is maximum.
...
Question 72:
Find two positive numbers \( x \) and \( y \) such that their sum is 35 and the product \( x^2 y^5 \) is maximum.
...
Question 73:
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
...
Question 74:
A square piece of tin of side \(18\,\text{cm}\) is to be made into a box without a top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box i...
Question 75:
A rectangular sheet of tin \(45\,\text{cm} \times 24\,\text{cm}\) is to be made into a box without a top, by cutting off a 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...