Mathematics · Textbook solutions

Application of Derivatives

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 143 questions

6.2 Rate of Change of Quantities

25 q

Solved Examples

Worked · 6
  1. 6.1 Eg.1
    Find the rate of change of the area of a circle per second with respect to its radius rr when r=5r = 5 cm.
  2. 6.1 Eg.2
    The volume of a cube is increasing at a rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of an edge is 10 centimetres?
  3. 6.1 Eg.3
    A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant, when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
  4. 6.1 Eg.4
    The length xx of a rectangle is decreasing at the rate of 3 cm/minute and the width yy is increasing at the rate of 2 cm/minute. When x=10x = 10 cm and y=6y = 6 cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.
  5. 6.1 Eg.5
    The total cost C(x)C(x) in Rupees, associated with the production of xx units of an item is given by C(x)=0.005x3−0.02x2+30x+5000C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000 Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.
  6. 6.1 Eg.6
    The total revenue in Rupees received from the sale of xx units of a product is given by R(x)=3x2+36x+5R(x) = 3x^2 + 36x + 5. Find the marginal revenue, when x=5x = 5, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.

Exercise 6.1

Practice · 19
  1. Find the rate of change of the area of a circle with respect to its radius rr when
    Ex 6.1 Q1(a)
    r=3r = 3 cm
  2. Ex 6.1 Q1(b)
    r=4r = 4 cm
  3. Ex 6.1 Q2
    The volume of a cube is increasing at the rate of 8 cm3/s8\ \text{cm}^3/\text{s}. How fast is the surface area increasing when the length of an edge is 12 cm?
  4. Ex 6.1 Q3
    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.
  5. Ex 6.1 Q4
    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?
  6. Ex 6.1 Q5
    A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
  7. Ex 6.1 Q6
    The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
  8. Ex 6.1 Q7
    The length xx of a rectangle is decreasing at the rate of 5 cm/minute and the width yy is increasing at the rate of 4 cm/minute. When x=8x = 8 cm and y=6y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
  9. Ex 6.1 Q8
    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.
  10. Ex 6.1 Q9
    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. Ex 6.1 Q10
    A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
  12. Ex 6.1 Q11
    A particle moves along the curve 6y=x3+26y = x^3 + 2. Find the points on the curve at which the yy-coordinate is changing 8 times as fast as the xx-coordinate.
  13. Ex 6.1 Q12
    The radius of an air bubble is increasing at the rate of 12\frac{1}{2} cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
  14. Ex 6.1 Q13
    A balloon, which always remains spherical, has a variable diameter 32(2x+1)\frac{3}{2}(2x + 1). Find the rate of change of its volume with respect to xx.
  15. Ex 6.1 Q14
    Sand is pouring from a pipe at the rate of 12 cm3/s12\ \text{cm}^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 sand cone increasing when the height is 4 cm?
  16. Ex 6.1 Q15
    The total cost C(x)C(x) in Rupees associated with the production of xx units of an item is given by C(x)=0.007x3−0.003x2+15x+4000C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000 Find the marginal cost when 17 units are produced.
  17. Ex 6.1 Q16
    The total revenue in Rupees received from the sale of xx units of a product is given by R(x)=13x2+26x+15R(x) = 13x^2 + 26x + 15 Find the marginal revenue when x=7x = 7.
  18. Ex 6.1 Q17
    The rate of change of the area of a circle with respect to its radius rr at r=6r = 6 cm is
    1. A.
      10π10\pi
    2. B.
      12π12\pi
    3. C.
      8π8\pi
    4. D.
      11π11\pi
  19. Ex 6.1 Q18
    The total revenue in Rupees received from the sale of xx units of a product is given by R(x)=3x2+36x+5R(x) = 3x^2 + 36x + 5. The marginal revenue, when x=15x = 15 is
    1. A.
      116
    2. B.
      96
    3. C.
      90
    4. D.
      126

6.3 Increasing and Decreasing Functions

30 q

Solved Examples

Worked · 7
  1. 6.2 Eg.7
    Show that the function given by f(x)=7x−3f(x) = 7x - 3 is increasing on R\mathbf{R}.
  2. 6.2 Eg.8
    Show that the function ff given by f(x)=x3−3x2+4x, x∈Rf(x) = x^3 - 3x^2 + 4x,\ x \in \mathbf{R} is increasing on R\mathbf{R}.
  3. 6.2 Eg.9
    Prove that the function given by f(x)=cos⁡xf(x) = \cos x is (a) decreasing in (0,π)(0, \pi) (b) increasing in (π,2π)(\pi, 2\pi), and (c) neither increasing nor decreasing in (0,2π)(0, 2\pi).
  4. 6.2 Eg.10
    Find the intervals in which the function ff given by f(x)=x2−4x+6f(x) = x^2 - 4x + 6 is (a) increasing (b) decreasing
  5. 6.2 Eg.11
    Find the intervals in which the function ff given by f(x)=4x3−6x2−72x+30f(x) = 4x^3 - 6x^2 - 72x + 30 is (a) increasing (b) decreasing.
  6. 6.2 Eg.12
    Find intervals in which the function given by f(x)=sin⁡3x, x∈[0,π2]f(x) = \sin 3x,\ x \in \left[0, \frac{\pi}{2}\right] is (a) increasing (b) decreasing.
  7. 6.2 Eg.13
    Find the intervals in which the function ff given by f(x)=sin⁡x+cos⁡x, 0≤x≤2πf(x) = \sin x + \cos x,\ 0 \le x \le 2\pi is increasing or decreasing.

Exercise 6.2

Practice · 23
  1. Ex 6.2 Q1
    Show that the function given by f(x)=3x+17f(x) = 3x + 17 is increasing on R\mathbf{R}.
  2. Ex 6.2 Q2
    Show that the function given by f(x)=e2xf(x) = e^{2x} is increasing on R\mathbf{R}.
  3. Ex 6.2 Q3
    Show that the function given by f(x)=sin⁡xf(x) = \sin x is (a) increasing in (0,π2)\left(0, \frac{\pi}{2}\right) (b) decreasing in (π2,π)\left(\frac{\pi}{2}, \pi\right) (c) neither increasing nor decreasing in (0,π)(0, \pi)
  4. Ex 6.2 Q4
    Find the intervals in which the function ff given by f(x)=2x2−3xf(x) = 2x^2 - 3x is (a) increasing (b) decreasing
  5. Ex 6.2 Q5
    Find the intervals in which the function ff given by f(x)=2x3−3x2−36x+7f(x) = 2x^3 - 3x^2 - 36x + 7 is (a) increasing (b) decreasing
  6. Find the intervals in which the following functions are strictly increasing or decreasing:
    Ex 6.2 Q6(a)
    x2+2x−5x^2 + 2x - 5
  7. Ex 6.2 Q6(b)
    10−6x−2x210 - 6x - 2x^2
  8. Ex 6.2 Q6(c)
    −2x3−9x2−12x+1-2x^3 - 9x^2 - 12x + 1
  9. Ex 6.2 Q6(d)
    6−9x−x26 - 9x - x^2
  10. Ex 6.2 Q6(e)
    (x+1)3(x−3)3(x + 1)^3 (x - 3)^3
  11. Ex 6.2 Q7
    Show that y=log⁡(1+x)−2x2+x, x>−1y = \log(1 + x) - \frac{2x}{2 + x},\ x > -1, is an increasing function of xx throughout its domain.
  12. Ex 6.2 Q8
    Find the values of xx for which y=[x(x−2)]2y = [x(x - 2)]^2 is an increasing function.
  13. Ex 6.2 Q9
    Prove that y=4sin⁡θ(2+cos⁡θ)−θy = \frac{4\sin\theta}{(2 + \cos\theta)} - \theta is an increasing function of θ\theta in [0,π2]\left[0, \frac{\pi}{2}\right].
  14. Ex 6.2 Q10
    Prove that the logarithmic function is increasing on (0,∞)(0, \infty).
  15. Ex 6.2 Q11
    Prove that the function ff given by f(x)=x2−x+1f(x) = x^2 - x + 1 is neither strictly increasing nor decreasing on (−1,1)(-1, 1).
  16. Ex 6.2 Q12
    Which of the following functions are decreasing on (0,π2)\left(0, \frac{\pi}{2}\right)?
    1. A.
      cos⁡x\cos x
    2. B.
      cos⁡2x\cos 2x
    3. C.
      cos⁡3x\cos 3x
    4. D.
      tan⁡x\tan x
  17. Ex 6.2 Q13
    On which of the following intervals is the function ff given by f(x)=x100+sin⁡x−1f(x) = x^{100} + \sin x - 1 decreasing?
    1. A.
      (0,1)(0, 1)
    2. B.
      (π2,π)\left(\frac{\pi}{2}, \pi\right)
    3. C.
      (0,π2)\left(0, \frac{\pi}{2}\right)
    4. D.
      None of these
  18. Ex 6.2 Q14
    For what values of aa the function ff given by f(x)=x2+ax+1f(x) = x^2 + ax + 1 is increasing on [1,2][1, 2]?
  19. Ex 6.2 Q15
    Let I be any interval disjoint from [−1,1][-1, 1]. Prove that the function ff given by f(x)=x+1xf(x) = x + \frac{1}{x} is increasing on I.
  20. Ex 6.2 Q16
    Prove that the function ff given by f(x)=log⁡sin⁡xf(x) = \log \sin x is increasing on (0,π2)\left(0, \frac{\pi}{2}\right) and decreasing on (π2,π)\left(\frac{\pi}{2}, \pi\right).
  21. Ex 6.2 Q17
    Prove that the function ff given by f(x)=log⁡∣cos⁡x∣f(x) = \log |\cos x| is decreasing on (0,π2)\left(0, \frac{\pi}{2}\right) and increasing on (3π2,2π)\left(\frac{3\pi}{2}, 2\pi\right).
  22. Ex 6.2 Q18
    Prove that the function given by f(x)=x3−3x2+3x−100f(x) = x^3 - 3x^2 + 3x - 100 is increasing in R\mathbf{R}.
  23. Ex 6.2 Q19
    The interval in which y=x2e−xy = x^2 e^{-x} is increasing is
    1. A.
      (−∞,∞)(-\infty, \infty)
    2. B.
      (−2,0)(-2, 0)
    3. C.
      (2,∞)(2, \infty)
    4. D.
      (0,2)(0, 2)

6.4 Maxima and Minima

64 q

Solved Examples

Worked · 16
  1. 6.3 Eg.14
    Find the maximum and the minimum values, if any, of the function ff given by f(x)=x2f(x) = x^2, x∈Rx \in \mathbf{R}.
  2. 6.3 Eg.15
    Find the maximum and minimum values of ff, if any, of the function given by f(x)=∣x∣f(x) = |x|, x∈Rx \in \mathbf{R}.
  3. 6.3 Eg.16
    Find the maximum and the minimum values, if any, of the function given by f(x)=xf(x) = x, x∈(0,1)x \in (0, 1).
  4. 6.3 Eg.17
    Find all points of local maxima and local minima of the function ff given by f(x)=x3−3x+3f(x) = x^3 - 3x + 3.
  5. 6.3 Eg.18
    Find all the points of local maxima and local minima of the function ff given by f(x)=2x3−6x2+6x+5f(x) = 2x^3 - 6x^2 + 6x + 5.
  6. 6.3 Eg.19
    Find local minimum value of the function ff given by f(x)=3+∣x∣f(x) = 3 + |x|, x∈Rx \in \mathbf{R}.
  7. 6.3 Eg.20
    Find local maximum and local minimum values of the function ff given by f(x)=3x4+4x3−12x2+12f(x) = 3x^4 + 4x^3 - 12x^2 + 12.
  8. [NCERT poses this same function as Example 18 and again as Example 21. Example 18 settles it with the FIRST derivative test; this one is worked to show that the SECOND derivative test FAILS here (f''(1) = 0) and one must fall back to the first. Both are kept because the two teach different things.]
    6.3 Eg.21
    Find all the points of local maxima and local minima of the function ff given by f(x)=2x3−6x2+6x+5f(x) = 2x^3 - 6x^2 + 6x + 5.
  9. 6.3 Eg.22
    Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.
  10. 6.3 Eg.23
    Find the shortest distance of the point (0,c)(0, c) from the parabola y=x2y = x^2, where 12≤c≤5\frac{1}{2} \le c \le 5.
  11. 6.3 Eg.24
    Let AP and BQ be two vertical poles at points A and B, respectively. If AP = 16 m, BQ = 22 m and AB = 20 m, then find the distance of a point R on AB from the point A such that RP2+RQ2\mathrm{RP}^2 + \mathrm{RQ}^2 is minimum.
  12. 6.3 Eg.25
    If length of three sides of a trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.
  13. 6.3 Eg.26
    Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
  14. 6.3 Eg.27
    Find the absolute maximum and minimum values of a function ff given by f(x)=2x3−15x2+36x+1f(x) = 2x^3 - 15x^2 + 36x + 1 on the interval [1,5][1, 5].
  15. 6.3 Eg.28
    Find absolute maximum and minimum values of a function ff given by f(x)=12x43−6x13f(x) = 12x^{\frac{4}{3}} - 6x^{\frac{1}{3}}, x∈[−1,1]x \in [-1, 1].
  16. 6.3 Eg.29
    An Apache helicopter of enemy is flying along the curve given by y=x2+7y = x^2 + 7. A soldier, placed at (3,7)(3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

Exercise 6.3

Practice · 48
  1. Find the maximum and minimum values, if any, of the following functions given by
    Ex 6.3 Q1(i)
    f(x)=(2x−1)2+3f(x) = (2x - 1)^2 + 3
  2. Ex 6.3 Q1(ii)
    f(x)=9x2+12x+2f(x) = 9x^2 + 12x + 2
  3. Ex 6.3 Q1(iii)
    f(x)=−(x−1)2+10f(x) = -(x - 1)^2 + 10
  4. Ex 6.3 Q1(iv)
    g(x)=x3+1g(x) = x^3 + 1
  5. Find the maximum and minimum values, if any, of the following functions given by
    Ex 6.3 Q2(i)
    f(x)=∣x+2∣−1f(x) = |x + 2| - 1
  6. Ex 6.3 Q2(ii)
    g(x)=−∣x+1∣+3g(x) = -|x + 1| + 3
  7. Ex 6.3 Q2(iii)
    h(x)=sin⁡(2x)+5h(x) = \sin(2x) + 5
  8. Ex 6.3 Q2(iv)
    f(x)=∣sin⁡4x+3∣f(x) = |\sin 4x + 3|
  9. Ex 6.3 Q2(v)
    h(x)=x+1h(x) = x + 1, x∈(−1,1)x \in (-1, 1)
  10. 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:
    Ex 6.3 Q3(i)
    f(x)=x2f(x) = x^2
  11. Ex 6.3 Q3(ii)
    g(x)=x3−3xg(x) = x^3 - 3x
  12. Ex 6.3 Q3(iii)
    h(x)=sin⁡x+cos⁡xh(x) = \sin x + \cos x, 0<x<π20 < x < \frac{\pi}{2}
  13. Ex 6.3 Q3(iv)
    f(x)=sin⁡x−cos⁡xf(x) = \sin x - \cos x, 0<x<2π0 < x < 2\pi
  14. Ex 6.3 Q3(v)
    f(x)=x3−6x2+9x+15f(x) = x^3 - 6x^2 + 9x + 15
  15. Ex 6.3 Q3(vi)
    g(x)=x2+2xg(x) = \frac{x}{2} + \frac{2}{x}, x>0x > 0
  16. Ex 6.3 Q3(vii)
    g(x)=1x2+2g(x) = \frac{1}{x^2 + 2}
  17. Ex 6.3 Q3(viii)
    f(x)=x1−xf(x) = x\sqrt{1 - x}, 0<x<10 < x < 1
  18. Prove that the following functions do not have maxima or minima:
    Ex 6.3 Q4(i)
    f(x)=exf(x) = e^x
  19. Ex 6.3 Q4(ii)
    g(x)=log⁡xg(x) = \log x
  20. Ex 6.3 Q4(iii)
    h(x)=x3+x2+x+1h(x) = x^3 + x^2 + x + 1
  21. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
    Ex 6.3 Q5(i)
    f(x)=x3f(x) = x^3, x∈[−2,2]x \in [-2, 2]
  22. Ex 6.3 Q5(ii)
    f(x)=sin⁡x+cos⁡xf(x) = \sin x + \cos x, x∈[0,π]x \in [0, \pi]
  23. Ex 6.3 Q5(iii)
    f(x)=4x−12x2f(x) = 4x - \frac{1}{2}x^2, x∈[−2,92]x \in \left[-2, \frac{9}{2}\right]
  24. Ex 6.3 Q5(iv)
    f(x)=(x−1)2+3f(x) = (x - 1)^2 + 3, x∈[−3,1]x \in [-3, 1]
  25. Ex 6.3 Q6
    Find the maximum profit that a company can make, if the profit function is given by p(x)=41−72x−18x2p(x) = 41 - 72x - 18x^2
  26. Ex 6.3 Q7
    Find both the maximum value and the minimum value of 3x4−8x3+12x2−48x+253x^4 - 8x^3 + 12x^2 - 48x + 25 on the interval [0,3][0, 3].
  27. Ex 6.3 Q8
    At what points in the interval [0,2π][0, 2\pi], does the function sin⁡2x\sin 2x attain its maximum value?
  28. Ex 6.3 Q9
    What is the maximum value of the function sin⁡x+cos⁡x\sin x + \cos x?
  29. Ex 6.3 Q10
    Find the maximum value of 2x3−24x+1072x^3 - 24x + 107 in the interval [1,3][1, 3]. Find the maximum value of the same function in [−3,−1][-3, -1].
  30. Ex 6.3 Q11
    It is given that at x=1x = 1, the function x4−62x2+ax+9x^4 - 62x^2 + ax + 9 attains its maximum value, on the interval [0,2][0, 2]. Find the value of aa.
  31. Ex 6.3 Q12
    Find the maximum and minimum values of x+sin⁡2xx + \sin 2x on [0,2π][0, 2\pi].
  32. Ex 6.3 Q13
    Find two numbers whose sum is 24 and whose product is as large as possible.
  33. Ex 6.3 Q14
    Find two positive numbers xx and yy such that x+y=60x + y = 60 and xy3xy^3 is maximum.
  34. Ex 6.3 Q15
    Find two positive numbers xx and yy such that their sum is 35 and the product x2y5x^2 y^5 is a maximum.
  35. Ex 6.3 Q16
    Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
  36. Ex 6.3 Q17
    A square piece of tin of side 18 cm is to be made into a box without 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 is the maximum possible.
  37. Ex 6.3 Q18
    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?
  38. Ex 6.3 Q19
    Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
  39. Ex 6.3 Q20
    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.
  40. Ex 6.3 Q21
    Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
  41. Ex 6.3 Q22
    A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
  42. Ex 6.3 Q23
    Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 827\frac{8}{27} of the volume of the sphere.
  43. Ex 6.3 Q24
    Show that the right circular cone of least curved surface and given volume has an altitude equal to 2\sqrt{2} time the radius of the base.
  44. Ex 6.3 Q25
    Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan⁡−12\tan^{-1}\sqrt{2}.
  45. Ex 6.3 Q26
    Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin⁡−1(13)\sin^{-1}\left(\frac{1}{3}\right).
  46. Ex 6.3 Q27
    The point on the curve x2=2yx^2 = 2y which is nearest to the point (0,5)(0, 5) is
    1. A.
      (22,4)(2\sqrt{2}, 4)
    2. B.
      (22,0)(2\sqrt{2}, 0)
    3. C.
      (0,0)(0, 0)
    4. D.
      (2,2)(2, 2)
  47. Ex 6.3 Q28
    For all real values of xx, the minimum value of 1−x+x21+x+x2\frac{1 - x + x^2}{1 + x + x^2} is
    1. A.
      00
    2. B.
      11
    3. C.
      33
    4. D.
      13\frac{1}{3}
  48. Ex 6.3 Q29
    The maximum value of [x(x−1)+1]13[x(x - 1) + 1]^{\frac{1}{3}}, 0≤x≤10 \le x \le 1 is
    1. A.
      (13)13\left(\frac{1}{3}\right)^{\frac{1}{3}}
    2. B.
      12\frac{1}{2}
    3. C.
      11
    4. D.
      00

Miscellaneous Exercise on Chapter 6

24 q

Solved Examples

Worked · 8
  1. Misc Eg.30
    A car starts from a point P at time t=0t = 0 seconds and stops at point Q. The distance xx, in metres, covered by it, in tt seconds is given by x=t2(2−t3)x = t^2\left(2 - \frac{t}{3}\right). Find the time taken by it to reach Q and also find distance between P and Q.
  2. Misc Eg.31
    A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan⁡−1(0.5)\tan^{-1}(0.5). Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m.
  3. Misc Eg.32
    A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres high. Find the rate at which the length of his shadow increases.
  4. Misc Eg.33
    Find intervals in which the function given by f(x)=310x4−45x3−3x2+365x+11f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 is (a) increasing (b) decreasing.
  5. Misc Eg.34
    Show that the function ff given by f(x)=tan⁡−1(sin⁡x+cos⁡x)f(x) = \tan^{-1}(\sin x + \cos x), x>0x > 0 is always an increasing function in (0,π4)\left(0, \frac{\pi}{4}\right).
  6. Misc Eg.35
    A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.
  7. Misc Eg.36
    An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.
  8. Misc Eg.37
    Manufacturer can sell xx items at a price of rupees (5−x100)\left(5 - \frac{x}{100}\right) each. The cost price of xx items is Rs (x5+500)\left(\frac{x}{5} + 500\right). Find the number of items he should sell to earn maximum profit.

Miscellaneous Exercise

Practice · 16
  1. Misc Q1
    Show that the function given by f(x)=log⁡xxf(x) = \frac{\log x}{x} has maximum at x=ex = e.
  2. Misc Q2
    The two equal sides of an isosceles triangle with fixed base bb are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?
  3. Misc Q3
    Find the intervals in which the function ff given by f(x)=4sin⁡x−2x−xcos⁡x2+cos⁡xf(x) = \frac{4\sin x - 2x - x\cos x}{2 + \cos x} is (i) increasing (ii) decreasing.
  4. Misc Q4
    Find the intervals in which the function ff given by f(x)=x3+1x3f(x) = x^3 + \frac{1}{x^3}, x≠0x \ne 0 is (i) increasing (ii) decreasing.
  5. Misc Q5
    Find the maximum area of an isosceles triangle inscribed in the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 with its vertex at one end of the major axis.
  6. Misc Q6
    A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3^3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
  7. Misc Q7
    The sum of the perimeter of a circle and square is kk, where kk is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
  8. Misc Q8
    A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
  9. Misc Q9
    A point on the hypotenuse of a triangle is at distance aa and bb from the sides of the triangle. Show that the minimum length of the hypotenuse is (a23+b23)32\left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right)^{\frac{3}{2}}.
  10. Misc Q10
    Find the points at which the function ff given by f(x)=(x−2)4(x+1)3f(x) = (x-2)^4(x+1)^3 has (i) local maxima (ii) local minima (iii) point of inflexion.
  11. Misc Q11
    Find the absolute maximum and minimum values of the function ff given by f(x)=cos⁡2x+sin⁡xf(x) = \cos^2 x + \sin x, x∈[0,π]x \in [0, \pi].
  12. Misc Q12
    Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius rr is 4r3\frac{4r}{3}.
  13. Misc Q13
    Let ff be a function defined on [a,b][a, b] such that f′(x)>0f'(x) > 0, for all x∈(a,b)x \in (a, b). Then prove that ff is an increasing function on (a,b)(a, b).
  14. Misc Q14
    Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R3\frac{2\mathrm{R}}{\sqrt{3}}. Also find the maximum volume.
  15. Misc Q15
    Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height hh and semi vertical angle α\alpha is one-third that of the cone and the greatest volume of cylinder is 427πh3tan⁡2α\frac{4}{27}\pi h^3 \tan^2\alpha.
  16. Misc Q16
    A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
    1. A.
      1 m/h
    2. B.
      0.1 m/h
    3. C.
      1.1 m/h
    4. D.
      0.5 m/h