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Mathematics · Class 12 Science

Ch 9Applications of Derivatives — Class 12 Mathematics, concept-first.

In the previous chapter we studied how to differentiate composite functions, inverse trigonometric functions, logarithmic functions and parametric functions, and we learned that the derivative at a point is precisely the slope of the tangent to the curve at that point.

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Key concepts

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Tangent and Normal to a Curve

The derivative dy/dx evaluated at a point on a curve gives the slope of the tangent line to the curve at that point — this is the geometric meaning of the derivative.

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Chapter contents

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Introduction

In the previous chapter we studied how to differentiate composite functions, inverse trigonometric functions, logarithmic functions and parametric functions, and we learned that the derivative at a po…

2.1.2

Application of Derivative in Geometry

Let be a continuous function of representing a curve in the -plane, and let be any point on the curve.

2.1.3

Derivative as a Rate measure

If is a given function, a change in from to is denoted , and the corresponding change in is . The ratio is called the average rate of change of with respect to over that interval; geometrically it is…

2.1.4

Velocity, Acceleration and Jerk

If is the displacement of a particle moving along a straight line, then is the rate of change of displacement with respect to time — in other words, is the particle's velocity.

2.2.1

Approximations

If is a differentiable function of , its derivative at is defined by the limit . Using the symbol for "approximately equal to", for a sufficiently small we have

2.3.1

Rolle's Theorem or Rolle's Lemma

Rolle's Theorem (or Rolle's Lemma): if a real-valued function is continuous on the closed interval , differentiable on the open interval , and , then there exists at least one point in the open interv…

2.3.2

Lagrange's Mean Value Theorem (LMVT)

Lagrange's Mean Value Theorem (LMVT): if a real-valued function is continuous on the closed interval and differentiable on the open interval , then there exists at least one point in such that

2.4.1

Increasing and decreasing functions

Increasing functions. A function is said to be monotonically (or strictly) increasing on an interval if, for any with , we have .

2.4.2

Maxima and Minima

Maxima of a function. A function is said to have a (local) maxima at if the value of the function at is greater than every other value of in a small neighbourhood of — that is, for a small and for eve…

2.4.3

First derivative test

First derivative test. A function has a maxima at if: (i) , (ii) [ is increasing for values of ], and (iii) [ is decreasing for values of ], where is a small positive number.

2.4.4

Second derivative test

Second derivative test. A function has a maxima at if and . A function has a minima at if and .

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 30 questions30 questions
  1. Q1The equation of tangent to the curve $y = x^2 + 4x + 1$ at $(-1, -2)$ is (a) $2x - y = 0$ (b) $2x + y - 5 = 0$ (c) $2x - y - 1 = 0$ (d) $x +…Preview
  2. Q2The displacement 's' of a moving particle at time 't' is given by $s = 5 + 20t - 2t^2$. Find its acceleration when the velocity is zero.Preview
  3. Q3A wire of length $l$ is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle…Preview
  4. Q4The equation of tangent to the curve $y = 3x^2 - x + 1$ at $P(1, 3)$ is (a) $5x - y = 2$ (b) $x + 5y = 16$ (c) $5x - y + 2 = 0$ (d) $5x = y$Preview
  5. Q5Test whether the function, $f(x) = x - \dfrac{1}{x}$, $x \in \mathbb{R}$, $x \ne 0$, is increasing or decreasing.Preview
  6. Q6A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of ₹3,000 per year from each subscriber. The…Preview
  7. Q7Find the maximum and minimum value of the function: $f(x) = 2x^3 - 21x^2 + 36x - 20$.Preview
  8. Q8Verify Rolle's theorem for the following function: $f(x) = x^2 - 4x + 10$ on $[0, 4]$.Preview
  9. Q9Find the equation of tangent to the curve $y = x^2 + 4x + 1$ at $(-1, -2)$.Preview
  10. Q10$f(x) = (x-1)(x-2)(x-3)$, $x \in [0,4]$, find 'c' if LMVT can be applied. **OR** A rod of 108 meters long is bent to form a rectangle. Find…Preview
  11. Q11The function $f(x) = x^x$ is minimum at $x = $ ________. (a) e (b) $-e$ (c) $\dfrac{1}{e}$ (d) $-\dfrac{1}{e}$Preview
  12. Q12The surface area of a spherical balloon is increasing at the rate of 2 cm$^2$/sec. At what rate the volume of the balloon is increasing when…Preview
  13. Q13Find the approximate value of $e^{1.005}$; given $e = 2.7183$.Preview
  14. Q14The maximum value of the function $f(x) = \dfrac{\log x}{x}$ is ________. (a) e (b) $\dfrac{1}{e}$ (c) $e^2$ (d) $\dfrac{1}{e^2}$Preview
  15. Q15Find the equation of tangent to the curve $y = 2x^3 - x^2 + 2$ at $\left(\dfrac{1}{2}, 2\right)$.Preview
  16. Q16Show that function $f(x) = \tan x$ is increasing in $\left(0, \dfrac{\pi}{2}\right)$.Preview
  17. Q17A wire of length 36 meters is bent to form a rectangle. Find its dimensions if the area of the rectangle is maximum.Preview
  18. Q18Show that the function $f(x) = x^3+10x+7$, $x \in R$ is strictly increasing.Preview
  19. Q19Find the approximate value of $\sin(30°30')$. Given that $1° = 0.0175^c$ and $\cos 30° = 0.866$Preview
  20. Q20Verify Lagrange's mean value theorem for the function $f(x) = \sqrt{x+4}$ on the interval $[0, 5]$.Preview
  21. Q21Find the approximate value of $\tan^{-1}(1.002)$. [Given: $\pi=3.1416$]Preview
  22. Q22A box with a square base is to have an open top. The surface area of box is 147 sq.cm. What should be its dimensions in order that the volum…Preview
  23. Q23Write the condition for the function $f(x)$, to be strictly increasing, for all $x\in R$.Preview
  24. Q24Divide the number 20 into two parts such that sum of their squares is minimum.Preview
  25. Q25The displacement of a particle at time $t$ is given by $s=2t^3-5t^2+4t-3$. Find the velocity and displacement at the time when the accelerat…Preview
  26. Q26Find the equations of tangent and normal to the curve $y=2x^3-x^2+2$ at point $\left(\dfrac12,2\right)$.Preview
  27. Q27The approximate value of the function $f(x)=x^3-3x+5$ at $x=1.99$ is ____. (a) 6.09 (b) 6.91 (c) 7.09 (d) 7.91Preview
  28. Q28Test whether the function $f(x)=x^3+6x^2+12x-7$ is increasing or decreasing for all $x\in R$.Preview
  29. Q29A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3…Preview
  30. Q30Verify LMVT for the function $f(x)=\log x$, on $[1,e]$.Preview

More questions

130 Q
+Show 23 questions23 questions
  1. Q1Find the equations of tangent and normal to the curve at the point on it: $y = x^2 + 2e^x + 2$ at $(0, 4)$.Free
  2. Q2Find the equations of tangent and normal to the curve at the point on it: $x^3 + y^3 - 9xy = 0$ at $(2, 4)$.Free
  3. Q3Find the equations of tangent and normal to the curve at the point on it: $x^2 - \sqrt3\,xy + 2y^2 = 5$ at $(\sqrt3, 2)$.Free
  4. Q4Find the equations of tangent and normal to the curve at the point on it: $2xy + \pi\sin y = 2\pi$ at $\left(1, \dfrac{\pi}{2}\right)$.Preview
  5. Q5Find the equations of tangent and normal to the curve at the point on it: $x\sin 2y = y\cos 2x$ at $\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\rig…Preview
  6. Q6Find the equations of tangent and normal to the curve at the point on it: $x = \sin\theta,\ y = \cos 2\theta$ at $\theta = \dfrac{\pi}{6}$.Preview
  7. Q7Find the equations of tangent and normal to the curve at the point on it: $x=\sqrt t,\ y = t - \dfrac{1}{\sqrt t}$ at $t=4$.Preview
  8. Q8Find the point on the curve $y = \sqrt{x-3}$ where the tangent is perpendicular to the line $6x + 3y - 5 = 0$.Preview
  9. Q9Find the points on the curve $y = x^3 - 2x^2 - x$ where the tangents are parallel to $3x - y + 1 = 0$.Preview
  10. Q10Find the equations of the tangents to the curve $x^2 + y^2 - 2x - 4y + 1 = 0$ which are parallel to the X-axis.Preview
  11. Q11Find the equations of the normals to the curve $3x^2 - y^2 = 8$, which are parallel to the line $x + 3y = 4$.Preview
  12. Q12If the line $y = 4x - 5$ touches the curve $y^2 = ax^3 + b$ at the point $(2, 3)$, find $a$ and $b$.Preview
  13. Q13A particle moves along the curve $6y = x^3 + 2$. Find the points on the curve at which the $y$-coordinate is changing $8$ times as fast as t…Preview
  14. Q14A spherical soap bubble is expanding so that its radius is increasing at the rate of $0.02$ cm/sec. At what rate is the surface area increas…Preview
  15. Q15The surface area of a spherical balloon is increasing at the rate of $2$ cm$^2$/sec. At what rate the volume of the balloon is increasing wh…Preview
  16. Q16If each side of an equilateral triangle increases at the rate of $\sqrt2$ cm/sec, find the rate of increase of its area when its side of len…Preview
  17. Q17The volume of a sphere increases at the rate of $20$ cm$^3$/sec. Find the rate of change of its surface area when its radius is $5$ cm.Preview
  18. Q18The edge of a cube is decreasing at the rate of $0.6$ cm/sec. Find the rate at which its volume is decreasing when the edge of the cube is $…Preview
  19. Q19A man of height 2 meters walks at a uniform speed of 6 km/hr away from a lamp post of 6 meters high. Find the rate at which the length of th…Preview
  20. Q20A man of height 1.5 meters walks toward a lamp post of height 4.5 meters, at the rate of $\dfrac34$ meter/sec. Find the rate at which (i) hi…Preview
  21. Q21A man of height 1.5 meters walks toward a lamp post of height 4.5 meters, at the rate of $\dfrac34$ meter/sec. Find the rate at which (ii) t…Preview
  22. Q22A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate…Preview
  23. Q23If water is poured into an inverted hollow cone whose semi-vertical angle is $30°$, so that its depth (measured along the axis) increases at…Preview
+Show 20 questions20 questions
  1. Q24Find the approximate value of $\sqrt{8.95}$.Free
  2. Q25Find the approximate value of $\sqrt{28}$.Free
  3. Q26Find the approximate value of $\sqrt{31.98}$.Free
  4. Q27Find the approximate value of $(3.97)^4$.Preview
  5. Q28Find the approximate value of $(4.01)^3$.Preview
  6. Q29Find the approximate value of $\sin(61°)$ given that $1° = 0.0175^c$, $\sqrt3 = 1.732$.Preview
  7. Q30Find the approximate value of $\sin(29°30')$ given that $1° = 0.0175^c$, $\sqrt3 = 1.732$.Preview
  8. Q31Find the approximate value of $\cos(60°30')$ given that $1° = 0.0175^c$, $\sqrt3 = 1.732$.Preview
  9. Q32Find the approximate value of $\tan(45°40')$ given that $1° = 0.0175^c$.Preview
  10. Q33Find the approximate value of $\tan^{-1}(0.999)$.Preview
  11. Q34Find the approximate value of $\cot^{-1}(0.999)$.Preview
  12. Q35Find the approximate value of $\tan^{-1}(1.001)$.Preview
  13. Q36Find the approximate value of $e^{0.995}$.Preview
  14. Q37Find the approximate value of $e^{2.1}$ given that $e^2 = 7.389$.Preview
  15. Q38Find the approximate value of $3^{2.01}$ given that $\log 3 = 1.0986$ (i.e. $\ln 3$).Preview
  16. Q39Find the approximate value of $\log_e(101)$ given that $\log_e 10 = 2.3026$.Preview
  17. Q40Find the approximate value of $\log_e(9.01)$ given that $\log 3 = 1.0986$ (i.e. $\ln 3$).Preview
  18. Q41Find the approximate value of $\log_{10}(1016)$ given that $\log_{10} e = 0.4343$.Preview
  19. Q42Find the approximate value of $f(x) = x^3 - 3x + 5$ at $x = 1.99$.Preview
  20. Q43Find the approximate value of $f(x) = x^3 + 5x^2 - 7x + 10$ at $x = 1.12$.Preview
+Show 18 questions18 questions
  1. Q44Check the validity of the Rolle's theorem for the function $f(x) = x^2 - 4x + 3,\ x \in [1, 3]$.Free
  2. Q45Check the validity of the Rolle's theorem for the function $f(x) = e^{-x}\sin x,\ x \in [0, \pi]$.Free
  3. Q46Check the validity of the Rolle's theorem for the function $f(x) = 2x^2 - 5x + 3,\ x \in [1, 3]$.Free
  4. Q47Check the validity of the Rolle's theorem for the function $f(x) = \sin x - \cos x + 3,\ x \in [0, 2\pi]$.Preview
  5. Q48Check the validity of the Rolle's theorem for the function $f(x) = x^2$ if $0 \le x \le 2$, $f(x) = 6-x$ if $2 \le x \le 6$.Preview
  6. Q49Check the validity of the Rolle's theorem for the function $f(x) = x^{2/3},\ x \in [-1, 1]$.Preview
  7. Q50Given an interval $[a, b]$ that satisfies hypothesis of Rolle's theorem for the function $f(x) = x^4 + x^2 - 2$. It is known that $a = -1$.…Preview
  8. Q51Verify Rolle's theorem for the function $f(x) = \sin x + \cos x + 7,\ x \in [0, 2\pi]$.Preview
  9. Q52Verify Rolle's theorem for the function $f(x) = \sin\dfrac{x}{2},\ x \in [0, 2\pi]$.Preview
  10. Q53Verify Rolle's theorem for the function $f(x) = x^2 - 5x + 9,\ x \in [1, 4]$.Preview
  11. Q54If Rolle's theorem holds for the function $f(x) = x^3 + px^2 + qx + 5,\ x \in [1, 3]$ with $c = 2 + \dfrac{1}{\sqrt3}$, find the values of $…Preview
  12. Q55Rolle's theorem holds for the function $f(x) = (x - 2)\log x,\ x \in [1, 2]$, show that the equation $x\log x = 2 - x$ is satisfied by at le…Preview
  13. Q56The function $f(x) = x(x+3)e^{-x/2}$ satisfies all the conditions of Rolle's theorem on $[-3, 0]$. Find the value of $c$ such that $f'(c) =…Preview
  14. Q57Verify Lagrange's mean value theorem for the function $f(x) = \log x$, on $[1, e]$.Preview
  15. Q58Verify Lagrange's mean value theorem for the function $f(x) = (x - 1)(x - 2)(x - 3)$ on $[0, 4]$.Preview
  16. Q59Verify Lagrange's mean value theorem for the function $f(x) = x^2 - 3x - 1,\ x \in \left[-\dfrac{11}{7}, \dfrac{13}{7}\right]$.Preview
  17. Q60Verify Lagrange's mean value theorem for the function $f(x) = 2x - x^2,\ x \in [0, 1]$.Preview
  18. Q61Verify Lagrange's mean value theorem for the function $f(x) = \dfrac{x-1}{x-3}$ on $[4, 5]$.Preview
+Show 38 questions38 questions
  1. Q62Test whether the following function is increasing or decreasing: $f(x) = x^3 - 6x^2 + 12x - 16,\ x \in \mathbb{R}$.Free
  2. Q63Test whether the following function is increasing or decreasing: $f(x) = 2 - 3x + 3x^2 - x^3,\ x \in \mathbb{R}$.Free
  3. Q64Test whether the following function is increasing or decreasing: $f(x) = x - \dfrac1x,\ x \in \mathbb{R},\ x \ne 0$.Free
  4. Q65Find the values of $x$ for which the function $f(x) = 2x^3 - 3x^2 - 12x + 6$ is strictly increasing.Preview
  5. Q66Find the values of $x$ for which the function $f(x) = 3 + 3x - 3x^2 + x^3$ is strictly increasing.Preview
  6. Q67Find the values of $x$ for which the function $f(x) = x^3 - 6x^2 - 36x + 7$ is strictly increasing.Preview
  7. Q68Find the values of $x$ for which the function $f(x) = 2x^3 - 3x^2 - 12x + 6$ is strictly decreasing.Preview
  8. Q69Find the values of $x$ for which the function $f(x) = x + \dfrac{25}{x}$ is strictly decreasing.Preview
  9. Q70Find the values of $x$ for which the function $f(x) = x^3 - 9x^2 + 24x + 12$ is strictly decreasing.Preview
  10. Q71Find the values of $x$ for which the function $f(x) = x^3 - 12x^2 - 144x + 13$ is (a) Increasing.Preview
  11. Q72Find the values of $x$ for which the function $f(x) = x^3 - 12x^2 - 144x + 13$ is (b) Decreasing.Preview
  12. Q73Find the values of $x$ for which $f(x) = 2x^3 - 15x^2 - 144x - 7$ is (a) strictly increasing.Preview
  13. Q74Find the values of $x$ for which $f(x) = 2x^3 - 15x^2 - 144x - 7$ is (b) strictly decreasing.Preview
  14. Q75Find the values of $x$ for which $f(x) = \dfrac{x}{x^2+1}$ is (a) strictly increasing.Preview
  15. Q76Find the values of $x$ for which $f(x) = \dfrac{x}{x^2+1}$ is (b) strictly decreasing.Preview
  16. Q77Show that $f(x) = 3x + \dfrac{1}{3x}$ increasing in $\left(\dfrac13, 1\right)$ and decreasing in $\left(\dfrac19, \dfrac13\right)$.Preview
  17. Q78Show that $f(x) = x - \cos x$ is increasing for all $x$.Preview
  18. Q79Find the maximum and minimum of the function $y = 5x^3 + 2x^2 - 3x$.Preview
  19. Q80Find the maximum and minimum of the function $f(x) = 2x^3 - 21x^2 + 36x - 20$.Preview
  20. Q81Find the maximum and minimum of the function $f(x) = x^3 - 9x^2 + 24x$.Preview
  21. Q82Find the maximum and minimum of the function $f(x) = x^2 + \dfrac{16}{x^2}$.Preview
  22. Q83Find the maximum and minimum of the function $f(x) = x\log x$.Preview
  23. Q84Find the maximum and minimum of the function $f(x) = \dfrac{\log x}{x}$.Preview
  24. Q85Divide the number 30 into two parts such that their product is maximum.Preview
  25. Q86Divide the number 20 into two parts such that sum of their squares is minimum.Preview
  26. Q87A wire of length 36 meter is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.Preview
  27. Q88A ball is thrown in the air. Its height at any time $t$ is given by $h = 3 + 14t - 5t^2$. Find the maximum height it can reach.Preview
  28. Q89Find the largest size of a rectangle that can be inscribed in a semi circle of radius 1 unit, so that two vertices lie on the diameter.Preview
  29. Q90An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of $\pi a^3$ cu. cm of water.…Preview
  30. Q91The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?Preview
  31. Q92A box with a square base is to have an open top. The surface area of the box is 192 sq.cm. What should be its dimensions in order that the v…Preview
  32. Q93The profit function $P(x)$ of a firm, selling $x$ items per day is given by $P(x) = (150 - x)x - 1625$. Find the number of items the firm sh…Preview
  33. Q94Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.Preview
  34. Q95Show that among rectangles of given area, the square has the least perimeter.Preview
  35. Q96Show that the height of a closed right circular cylinder, of a given volume and least surface area, is equal to its diameter.Preview
  36. Q97Find the volume of the largest cylinder that can be inscribed in a sphere of radius $r$ cm.Preview
  37. Q98Show that $y = \log(1+x) - \dfrac{2x}{2+x},\ x > -1$ is an increasing function on its domain.Preview
  38. Q99Prove that $y = \dfrac{4\sin\theta}{2+\cos\theta} - \theta$ is an increasing function of $\theta \in \left[0, \dfrac{\pi}{2}\right]$.Preview
+Show 10 questions10 questions
  1. Q100If the function $f(x) = ax^3 + bx^2 + 11x - 6$ satisfies conditions of Rolle's theorem in $[1, 3]$ and $f'\left(2 + \dfrac{1}{\sqrt3}\right)…Free
  2. Q101If $f(x) = \dfrac{x^2 - 1}{x^2 + 1}$, for every real $x$, then the minimum value of $f$ is - (A) 1 (B) 0 (C) $-1$ (D) 2Free
  3. Q102A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate…Free
  4. Q103Let $f(x)$ and $g(x)$ be differentiable for $0 < x < 1$ such $f(0) = 0, g(0) = 0, f(1) = 6$. Let there exist a real number $c$ in $(0, 1)$ s…Preview
  5. Q104If $f(x) = x^3 - 6x^2 + 9x + 18$, then $f(x)$ is strictly decreasing in - (A) $(-\infty, 1)$ (B) $[3, \infty)$ (C) $(-\infty, 1] \cup [3, \i…Preview
  6. Q105If $x = -1$ and $x = 2$ are the extreme points of $y = \alpha\log x + \beta x^2 + x$ then (A) $\alpha=-6,\beta=\dfrac12$ (B) $\alpha=-6,\bet…Preview
  7. Q106The normal to the curve $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$ (A) Meets the curve again in second quadrant. (B) Does not meet the curve again.…Preview
  8. Q107The equation of the tangent to the curve $y = 1 - e^{x/2}$ at the point of intersection with Y-axis is (A) $x + 2y = 0$ (B) $2x + y = 0$ (C)…Preview
  9. Q108If the tangent at $(1, 1)$ on $y^2 = x(2 - x)^2$ meets the curve again at $P$ then $P$ is (A) $(4, 4)$ (B) $(-1, 2)$ (C) $(3, 6)$ (D) $\left…Preview
  10. Q109The approximate value of $\tan(44°30')$ given that $1° = 0.0175$. (A) 0.8952 (B) 0.9528 (C) 0.9285 (D) 0.9825Preview
+Show 21 questions21 questions
  1. Q110If the curves $ax^2 + by^2 = 1$ and $a'x^2 + b'y^2 = 1$ intersect orthogonally, then prove that $\dfrac{1}{a} - \dfrac{1}{b} = \dfrac{1}{a'}…Free
  2. Q111Determine the area of the triangle formed by the tangent to the graph of the function $y = 3 - x^2$ drawn at the point $(1, 2)$ and the coor…Free
  3. Q112Find the equation of the tangent and normal drawn to the curve $y^4 - 4x^4 - 6xy = 0$ at the point $M(1, 2)$.Free
  4. Q113A water tank in the form of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and t…Preview
  5. Q114Find all points on the ellipse $9x^2 + 16y^2 = 400$, at which the $y$-coordinate is decreasing and the $x$-coordinate is increasing at the s…Preview
  6. Q115Verify Rolle's theorem for the function $f(x) = \dfrac{2}{e^x + e^{-x}}$ on $[-1, 1]$.Preview
  7. Q116The position of a particle is given by the function $s(t) = 2t^2 + 3t - 4$. Find the time $t = c$ in the interval $0 \le t \le 4$ when the i…Preview
  8. Q117Find the approximate value of the function $f(x) = \sqrt{x^2 + 3x}$ at $x = 1.02$.Preview
  9. Q118Find the approximate value of $\cos^{-1}(0.51)$ given $\pi = 3.1416$, $\dfrac{2}{\sqrt3} = 1.1547$.Preview
  10. Q119Find the intervals on which the function $y = x^x,\ (x > 0)$ is increasing and decreasing.Preview
  11. Q120Find the intervals on the which the function $f(x) = \dfrac{x}{\log x}$, is increasing and decreasing.Preview
  12. Q121An open box with a square base is to be made out of a given quantity of sheet of area $a^2$. Show the maximum volume of the box is $\dfrac{a…Preview
  13. Q122Show that of all rectangles inscribed in a given circle, the square has the maximum area.Preview
  14. Q123Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.Preview
  15. Q124A window is in the form of a rectangle surmounted by a semi-circle. If the perimeter be 30 m, find the dimensions so that the greatest possi…Preview
  16. Q125Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of…Preview
  17. Q126A wire of length $l$ is cut in to two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of t…Preview
  18. Q127A rectangular sheet of paper of fixed perimeter with the sides having their length in the ratio $8 : 15$ converted in to an open rectangular…Preview
  19. Q128Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\dfrac{4r}{3}$.Preview
  20. Q129Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\dfrac{2R}{\sqrt3}$. Also find th…Preview
  21. Q130Find the maximum and minimum values of the function $f(x) = \cos^2 x + \sin x$.Preview