JEE Main - Applications: rate of change monotonicity maxima and minima Online Test (MCQ Quiz)

1. If y = x^3 - 6x^2 + 5, find the interval where the function is strictly increasing.

A) (0, 2) B) (-inf, 0) U (4, inf) C) (2, 4) D) (-inf, inf)

2. Find the point on the curve y = x^2 that is closest to the point (2, 0).

A) (1, 1) B) (2, 4) C) (0, 0) D) (1, -1)

3. The rate of change of the volume of a sphere with respect to its radius r is given by:

A) 4*pi*r B) 4*pi*r^2 C) 4/3*pi*r^3 D) 8*pi*r

4. Find the local maximum value of the function f(x) = x^3 - 3x + 1.

A) 1 B) 3 C) -1 D) 0

5. If the side length of a square is increasing at a rate of 0.5 cm/s, what is the rate of change of its area when the side length is 4 cm?

A) 2 cm^2/s B) 4 cm^2/s C) 8 cm^2/s D) 1 cm^2/s

6. Determine the interval where the function f(x) = sin(x) is decreasing in the interval [0, 2*pi].

A) (0, pi) B) (pi, 2*pi) C) (pi/2, 3*pi/2) D) (0, pi/2)

7. Find the absolute maximum value of f(x) = x^3 on the interval [-1, 2].

A) -1 B) 8 C) 1 D) 0

8. The volume of a cube is increasing at a rate of 10 cm^3/min. How fast is the surface area of the cube increasing when the edge length is 5 cm?

A) 4 cm^2/min B) 8 cm^2/min C) 10 cm^2/min D) 2 cm^2/min

9. Find the value of x for which the function f(x) = 2x^3 - 9x^2 + 12x + 1 has a local maximum.

A) x = 1 B) x = 2 C) x = 3 D) x = 0

10. A ladder 10 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.1 m/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 m from the wall?

A) 0.075 m/s B) 0.15 m/s C) 0.05 m/s D) 0.1 m/s

11. Find the interval of concavity for the function f(x) = x^4 - 4x^3.

A) (-inf, 0) B) (0, inf) C) (-inf, 0) U (0, inf) D) (0, 2)

12. The radius of a circle is increasing at a rate of 2 cm/s. What is the rate of change of its area when the radius is 5 cm?

A) 10*pi cm^2/s B) 20*pi cm^2/s C) 5*pi cm^2/s D) 25*pi cm^2/s

13. Find the local minimum value of the function f(x) = x^3 - 6x^2 + 5.

A) 5 B) -27 C) -11 D) 0

14. Water is leaking out of a conical tank at a rate of 3 m^3/min. The tank has a height of 10 m and a radius of 5 m. How fast is the water level falling when the water is 8 m deep?

A) 15/(16*pi) m/min B) 3/(4*pi) m/min C) 5/(8*pi) m/min D) 3/(16*pi) m/min

15. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 12x - 5.

A) (2, 3) B) (1, 2) C) (2, 5) D) (3, 4)

16. If the distance traveled by an object is given by s(t) = t^3 - 6t^2 + 5t, find the time when the velocity is zero.

A) t=1, t=5 B) t=2, t=3 C) t=0, t=6 D) t=1, t=6

17. Find the absolute minimum value of f(x) = x^4 - 2x^2 + 3 on the interval [-2, 2].

A) 3 B) 1 C) 2 D) 0

18. The area of a rectangle is increasing at a rate of 5 cm^2/s. If the length of the rectangle is kept constant at 10 cm, at what rate is the width changing?

A) 0.5 cm/s B) 0.25 cm/s C) 1 cm/s D) 0.1 cm/s

19. Find the interval where the function f(x) = x^4 - 2x^2 is concave up.

A) (-1, 1) B) (-inf, -1) U (1, inf) C) (-inf, 0) D) (0, inf)

20. A particle moves along the curve y = x^3. If the x-coordinate is increasing at a rate of 2 units/sec, at what rate is the y-coordinate changing when x = 3?

A) 6 units/sec B) 18 units/sec C) 54 units/sec D) 27 units/sec

21. Find the local maximum value of f(x) = -x^3 + 3x^2 - 5.

A) -5 B) -7 C) -3 D) -2

22. The volume of a cylinder is increasing at a rate of 10 cm^3/sec. If the radius is fixed at 5 cm, how fast is the height changing?

A) 10/(25*pi) cm/sec B) 10/(5*pi) cm/sec C) 10/(50*pi) cm/sec D) 10/(pi) cm/sec

23. Find the point of local minimum for the function f(x) = x^3 - 12x + 1.

A) (2, -15) B) (-2, 17) C) (2, 17) D) (-2, -15)

24. A car is traveling at 30 m/s when the brakes are applied. The distance it travels before stopping is given by s(t) = 30t - 5t^2. Find the time it takes to stop.

A) 2 seconds B) 3 seconds C) 5 seconds D) 6 seconds

25. Find the interval where the function f(x) = x^5 - 5x^4 is strictly decreasing.

A) (0, 4) B) (-inf, 0) U (4, inf) C) (4, inf) D) (-inf, 0)

26. The tip of the minute hand of a clock is 10 cm from the center. How fast is the tip moving when the time is 3:00 PM?

A) 0 cm/min B) 5*pi/3 cm/min C) 10*pi/3 cm/min D) 20*pi/3 cm/min

27. Find the absolute maximum value of f(x) = 3x^4 - 4x^3 on the interval [-1, 2].

A) 16 B) 0 C) 7 D) -7/3

28. If the perimeter of a square is increasing at a rate of 4 cm/s, at what rate is the area increasing when the side length is 10 cm?

A) 40 cm^2/s B) 80 cm^2/s C) 20 cm^2/s D) 10 cm^2/s

29. Find the value of x where f(x) = x^3 - 3x^2 + 4 has a local minimum.

A) x = 0 B) x = 1 C) x = 2 D) x = 3

30. Two cars start moving from the same point. One travels north at 30 km/h and the other travels east at 40 km/h. At what rate is the distance between them increasing after 2 hours?

A) 50 km/h B) 60 km/h C) 70 km/h D) 80 km/h

31. Find the interval of concavity for the function f(x) = sin(x) in the interval [0, 2*pi].

A) (0, pi) B) (pi, 2*pi) C) (pi/2, 3*pi/2) D) (0, 2*pi)

32. The height of a triangle is increasing at a rate of 2 cm/min, while the base is decreasing at a rate of 3 cm/min. Find the rate of change of the area when the height is 10 cm and the base is 5 cm.

A) -5 cm^2/min B) 5 cm^2/min C) -10 cm^2/min D) 10 cm^2/min

33. Find the local extrema of f(x) = x^2 * e^x.

A) x = 0 (min), x = -2 (max) B) x = 0 (max), x = -2 (min) C) x = 0 (min), x = 2 (max) D) x = 0 (max), x = 2 (min)

34. A balloon is being inflated at a rate of 100 cm^3/sec. How fast is the radius of the balloon increasing when the radius is 10 cm?

A) 1/(10*pi) cm/sec B) 1/(2*pi) cm/sec C) 1/(20*pi) cm/sec D) 1/(4*pi) cm/sec

35. Find the absolute minimum value of f(x) = x^3 - 3x + 1 on the interval [-2, 3].

A) -1 B) 1 C) -2 D) 19

36. The length of a rectangle is decreasing at a rate of 2 cm/s and the width is increasing at a rate of 3 cm/s. Find the rate of change of the area when the length is 10 cm and the width is 5 cm.

A) 5 cm^2/s B) -5 cm^2/s C) 10 cm^2/s D) -10 cm^2/s

37. Find the interval where the function f(x) = x/(x^2 + 1) is increasing.

A) (-1, 1) B) (-inf, -1) U (1, inf) C) (-inf, 0) D) (0, inf)

38. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

A) 1/8 m/s B) 1/10 m/s C) 8/10 m/s D) 10/8 m/s

39. Find the local maximum value of f(x) = x*e^(-x^2).

A) 1/sqrt(2e) B) 0 C) -1/sqrt(2e) D) e

40. The radius of a sphere is increasing at a rate of 0.1 cm/s. How fast is the volume changing when the radius is 5 cm?

A) 10*pi cm^3/s B) 2.5*pi cm^3/s C) 5*pi cm^3/s D) 20*pi cm^3/s

41. Find the interval where the function f(x) = x^3 - 6x^2 + 5 is decreasing.

A) (0, 4) B) (-inf, 0) U (4, inf) C) (0, 2) D) (2, 4)

42. Find the point of inflection for f(x) = x^4 - 6x^2.

A) (sqrt(3), -9) and (-sqrt(3), -9) B) (0, 0) C) (sqrt(3), 0) and (-sqrt(3), 0) D) (3, -27) and (-3, -27)

43. A rectangular field is to be enclosed by a fence. If 1000 m of fencing is available, find the dimensions of the field that will maximize the area.

A) 250m x 250m B) 200m x 300m C) 100m x 400m D) 500m x 500m

44. The rate of change of population is proportional to the current population P. If dP/dt = kP, find the population after time t, given P(0) = P0.

A) P(t) = P0 * e^(kt) B) P(t) = P0 + kt C) P(t) = P0 * (1 + kt) D) P(t) = P0 / (1 + kt)

45. Find the absolute maximum value of f(x) = x^3 - 3x^2 + 5 on the interval [0, 3].

A) 5 B) 0 C) 2 D) -1

46. The area of a circle is increasing at a rate of 10 cm^2/min. Find the rate at which the radius is increasing when the area is 25*pi cm^2.

A) 1/pi cm/min B) 2/pi cm/min C) 1/(2*pi) cm/min D) 10/pi cm/min

47. Find the local minimum value of f(x) = x^4 - 4x.

A) -3 B) 0 C) 3 D) 4

48. A particle's position is given by s(t) = t^3 - 9t^2 + 15t. Find the time intervals when the particle is moving to the left.

A) (1, 5) B) (-inf, 1) U (5, inf) C) (1, inf) D) (-inf, 1)

49. Find the interval where the function f(x) = x^2 * ln(x) is concave down.

A) (0, 1/sqrt(e)) B) (0, sqrt(e)) C) (1/sqrt(e), inf) D) (0, 1)

50. A cylindrical can is to have a volume of 1000 cm^3. Find the dimensions (radius and height) that minimize the surface area.

A) r = 10/cbrt(pi), h = 10*cbrt(pi) B) r = 5*cbrt(2/pi), h = 5*cbrt(2/pi) C) r = 10/sqrt(pi), h = 10*sqrt(pi) D) r = 5, h = 40/pi

Applications: rate of change monotonicity maxima and minima - Online Test

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