JEE (Main) Mathematics - Rate of Change, Maxima and Minima MCQs

46. The rate of change of a quantity is defined as:
    A) The derivative of the quantity with respect to time
    B) The slope of the tangent to the curve
    C) The second derivative of the quantity
    D) The integral of the quantity
    Answer: A) The derivative of the quantity with respect to time
47. If a function f(x) is increasing, then:
    A) f'(x) < 0 for all x
    B) f'(x) > 0 for all x
    C) f'(x) = 0 for all x
    D) f'(x) changes sign
    Answer: B) f'(x) > 0 for all x
48. If a function f(x) is decreasing, then:
    A) f'(x) < 0 for all x
    B) f'(x) > 0 for all x
    C) f'(x) = 0 for all x
    D) f'(x) changes sign
    Answer: A) f'(x) < 0 for all x
49. The function f(x) = x² has:
    A) A local maximum at x = 0
    B) A local minimum at x = 0
    C) A local maximum at x = 1
    D) No local extremum
    Answer: B) A local minimum at x = 0
50. The first derivative test is used to find:
    A) The rate of change of a function
    B) The maxima and minima of a function
    C) The concavity of a function
    D) The area under the curve
    Answer: B) The maxima and minima of a function
51. For a function f(x), if f'(x) > 0 and f''(x) > 0, then:
    A) f(x) is increasing and concave up
    B) f(x) is decreasing and concave up
    C) f(x) is increasing and concave down
    D) f(x) is decreasing and concave down
    Answer: A) f(x) is increasing and concave up
52. The function f(x) = -x² + 4x has:
    A) A local maximum at x = 0
    B) A local minimum at x = 0
    C) A local maximum at x = 2
    D) A local minimum at x = 2
    Answer: C) A local maximum at x = 2
53. To find the critical points of a function, we compute:
    A) f'(x) = 0 and f''(x) = 0
    B) f'(x) = 0 and solve for x
    C) f'(x) = 0 and analyze concavity
    D) None of the above
    Answer: B) f'(x) = 0 and solve for x
54. The function f(x) = x³ - 3x² + 4 has:
    A) A local maximum at x = 1
    B) A local minimum at x = 1
    C) A point of inflection at x = 1
    D) No critical points
    Answer: B) A local minimum at x = 1
55. At a local maximum, the first derivative of the function is:
    A) Positive
    B) Negative
    C) Zero
    D) Undefined
    Answer: C) Zero
56. The second derivative test is used to:
    A) Find the value of the function
    B) Determine whether a critical point is a maxima or minima
    C) Calculate the slope of the function
    D) Find the integral of the function
    Answer: B) Determine whether a critical point is a maxima or minima
57. If the second derivative of a function f(x) at a point x = a is positive, then:
    A) f(x) has a local maximum at x = a
    B) f(x) has a local minimum at x = a
    C) f(x) is concave up at x = a
    D) f(x) is concave down at x = a
    Answer: B) f(x) has a local minimum at x = a
58. For the function f(x) = x⁴ - 4x², the local extrema are:
    A) Local minimum at x = 0, Local maximum at x = ±2
    B) Local maximum at x = 0, Local minimum at x = ±2
    C) Local maximum at x = ±2
    D) No local extrema
    Answer: A) Local minimum at x = 0, Local maximum at x = ±2
59. The rate of change of a function f(x) at a point x = a is given by:
    A) f'(a)
    B) f(a)
    C) f''(a)
    D) f(a) - f(b)
    Answer: A) f'(a)
60. A function f(x) is said to be monotonic if:
    A) It has no turning points
    B) Its derivative does not change sign
    C) It is continuous
    D) It has both local maxima and minima
    Answer: B) Its derivative does not change sign

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