JEE Mathematics Monotonic Functions and Lagranges Theorem MCQs

Choose the most appropriate option (a, b, c or d).

Question. If \( f(x) = x^3 + 4x^2 + \lambda x + 1 \) is a monotonically decreasing function of x in the largest possible interval \( (-2, -2/3) \) then
(a) \( \lambda = 4 \)
(b) \( \lambda = 2 \)
(c) \( \lambda = -1 \)
(d) \( \lambda \) has no real value
Answer: (a) \( \lambda = 4 \)

Question. The function \( f(x) = \sin^4 x + \cos^4 x \) increases if
(a) \( 0 < x < \frac{\pi}{8} \)
(b) \( \frac{\pi}{4} < x < \frac{3\pi}{8} \)
(c) \( \frac{3\pi}{4} < x < \frac{5\pi}{8} \)
(d) \( \frac{5\pi}{8} < x < \frac{3\pi}{4} \)
Answer: (b) \( \frac{\pi}{4} < x < \frac{3\pi}{8} \)

Question. If \( f(x) = \frac{x}{\sin x} \) and \( g(x) = \frac{x}{\tan x} \), where \( 0 < x \leq 1 \), then in the interval
(a) both f(x) and g(x) are increasing functions
(b) both f(x) and g(x) are decreasing functions
(c) f(x) is an increasing function
(d) g(x) is an increasing function
Answer: (c) f(x) is an increasing function

Question. The function \( f(x) = \tan^{-1} x - x \) is monotonically decreasing in the set
(a) R
(b) \( (0, +\infty) \)
(c) \( R - \{0\} \)
(d) None of the options
Answer: (c) \( R - \{0\} \)

Question. Let \( f(x) = \tan^{-1}\{\phi(x)\} \), where \( \phi(x) \) is m.i. for \( 0 < x < \pi/2 \). Then f(x) is
(a) increasing in \( (0, \pi/2) \)
(b) decreasing in \( (0, \pi/2) \)
(c) increasing in \( (0, \pi/4) \) and decreasing in \( (\pi/4, \pi/2) \)
(d) None of the options
Answer: (a) increasing in \( (0, \pi/2) \)

Question. If \( f(x) = \frac{p^2 - 1}{p^2 + 1} x^3 - 3x + \log 2 \) is a decreasing function of x in R then the set of possible values of p (independent of x) is
(a) [-1, 1]
(b) [1, \( \infty \))
(c) (-\( \infty \), -1]
(d) None of the options
Answer: (a) [-1, 1]

Question. If \( f(x) = (ab - b^2 - 2)x + \int_0^x (\cos^4 \theta + \sin^4 \theta) d\theta \) is a decreasing function of x for all \( x \in R \) and \( b \in R \), b being independent of x, then
(a) \( a \in (0, \sqrt{6}) \)
(b) \( a \in (-\sqrt{6}, \sqrt{6}) \)
(c) \( a \in (-\sqrt{6}, 0) \)
(d) None of the options
Answer: (b) \( a \in (-\sqrt{6}, \sqrt{6}) \)

Question. Let \( f(x) = \cos \pi x + 10x + 3x^2 + x^3 \), \( -2 \leq x \leq 3 \). The absolute minimum value of f(x) is
(a) 0
(b) -15
(c) \( 3 - 2\pi \)
(d) None of the options
Answer: (b) -15

Question. If \( x \in [-1, 1] \) then the minimum value of \( f(x) = x^2 + x + 1 \) is
(a) -3/4
(b) -15
(c) \( 3 - 2\pi \)
(d) None of the options
Answer: (a) -3/4

Question. Let \( f(x) = \frac{4}{3} x^3 - 4x \), \( 0 \leq x \leq 2 \). Then the global minimum value of the function is
(a) 0
(b) -8/3
(c) -4
(d) None of the options
Answer: (b) -8/3

Question. Let \( f(x) = 6 - 12x + 9x^2 - 2x^3 \), \( 1 \leq x \leq 4 \). Then the absolute maximum value of f(x) in the interval is
(a) 2
(b) 1
(c) 4
(d) None of the options
Answer: (b) 1

Question. If \( f(x) = \int_0^x (t^2 + 2t + 2)dt \), \( 2 \leq x \leq 4 \), then
(a) the maximum value of f(x) is \( \frac{136}{3} \)
(b) the minimum value of f(x) is 10
(c) the maximum value of f(x) is 26
(d) None of the options
Answer: (a) the maximum value of f(x) is \( \frac{136}{3} \)

Question. Let the interval I = [-1, 4] and f : I \( \rightarrow \) R be a function such that \( f(x) = x^3 - 3x \). Then the range of the function is
(a) [2, 52]
(b) [-2, 2]
(c) [-2, 52]
(d) None of the options
Answer: (c) [-2, 52]

Question. The range of the function \( f(x) = |2x + 1| - 2|x - 1| \), \( x \in R \), is
(a) [-3, 3]
(b) [0, 6]
(c) R
(d) None of the options
Answer: (a) [-3, 3]

Question. Let the function f(x) be defined as follows :
\( f(x) = x^3 + x^2 - 10x, -1 \leq x < 0 \)
\( \cos x, 0 \leq x < \frac{\pi}{2} \)
\( 1 + \sin x, \frac{\pi}{2} \leq x \leq \pi. \)
Then f(x) is
(a) a local minimum at \( x = \pi/2 \)
(b) a local maximum at \( x = \pi/2 \)
(c) an absolute minimum at x = -1
(d) an absolute maximum at x = \( \pi \)
Answer: (b) a local maximum at \( x = \pi/2 \)

Question. Let f(x) be a function defined as below :
\( f(x) = \sin(x^2 - 3x), x \leq 0 \)
\( 6x + 5x^2, x > 0 \)
Then at x = 0, f(x)
(a) has a local maximum
(b) has a local minimum
(c) is discontinuous
(d) None of the options
Answer: (b) has a local minimum

Question. If \( \theta \) is a positive acute angle then
(a) \( \tan \theta < \theta < \sin \theta \)
(b) \( \theta < \sin \theta < \tan \theta \)
(c) \( \sin \theta < \tan \theta < \theta \)
(d) None of the options
Answer: (d) None of the options

Question. Let f : R \( \rightarrow \) R be a function such that \( f(x) = ax + 3 \sin x + 4 \cos x \). Then f(x) is invertible if
(a) \( a \in (-5, 5) \)
(b) \( a \in (-\infty, -5) \)
(c) \( a \in (5, +\infty) \)
(d) None of the options
Answer: (a) \( a \in (-5, 5) \)

Question. The function \( f(x) = x^3 + \lambda x^2 + 5x + \sin 2x \) will be an invertible function if \( \lambda \) belongs to
(a) \( (-\infty, -3) \)
(b) (-3, 3)
(c) \( (3, +\infty) \)
(d) None of the options
Answer: (b) (-3, 3)

Question. The value of c in Lagrange’s theorem for the function |x| in the interval [-1, 1] is
(a) 0
(b) 1/2
(c) -1/2
(d) nonexistent in the interval
Answer: (d) nonexistent in the interval

Question. The equation \( \sin x + x \cos x = 0 \) has at least one root in the interval
(a) \( (-\pi/2, 0) \)
(b) \( (0, \pi) \)
(c) \( (-\pi/2, \pi/2) \)
(d) None of the options
Answer: (b) \( (0, \pi) \)

Question. If \( 4a + 2b + c = 0 \) then the equation \( 3ax^2 + 2bx + c = 0 \) has at least one real root lying between
(a) 0 and 1
(b) 1 and 2
(c) 0 and 2
(d) None of the options
Answer: (c) 0 and 2

Question. If the equation \( a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x = 0, a_1 \neq 0, n \geq 2 \), has a positive root \( \alpha \) then the equation \( n a_n x^{n-1} + (n - 1) a_{n-1} x^{n-2} + \dots + a_1 = 0 \) has a positive root which is
(a) greater than \( \alpha \)
(b) smaller than \( \alpha \)
(c) greater than or equal to \( \alpha \)
(d) equal to \( \alpha \)
Answer: (b) smaller than \( \alpha \)

Choose the correct options. One or more options may be correct.

Question. Let \( h(x) = f(x) - \{f(x)\}^2 + \{f(x)\}^3 \) for all real values of x. Then
(a) h is increasing whenever f(x) is increasing
(b) h is increasing whenever f'(x) < 0
(c) h is decreasing whenever f is decreasing
(d) nothing can be said in general
Answer: (a) h is increasing whenever f(x) is increasing, (c) h is decreasing whenever f is decreasing

Question. If \( f(x) = \sin x, -\pi/2 \leq x \leq \pi/2 \), then
(a) f(x) is increasing in the interval \( [-\pi/2, \pi/2] \)
(b) f{f(x)} is increasing in the interval \( [-\pi/2, \pi/2] \)
(c) f{f(x)} is decreasing in \( [-\pi/2, 0] \) and increasing in \( [0, \pi/2] \)
(d) f{f(x)} is invertible in \( [-\pi/2, \pi/2] \)
Answer: (a) f(x) is increasing in the interval \( [-\pi/2, \pi/2] \), (b) f{f(x)} is increasing in the interval \( [-\pi/2, \pi/2] \), (d) f{f(x)} is invertible in \( [-\pi/2, \pi/2] \)

Question. Let \( f(x) = 2 \sin^3 x - 3 \sin^2 x + 12 \sin x + 5, 0 \leq x \leq \pi/2 \). Then f(x) is
(a) decreasing in \( [0, \pi/2] \)
(b) increasing in \( [0, \pi/2] \)
(c) increasing in \( [0, \pi/4] \) and decreasing in \( [\pi/4, \pi/2] \)
(d) None of the options
Answer: (b) increasing in \( [0, \pi/2] \)

Question. Let \( f(x) = -2x^3 + 21x^2 - 60x + 41 \). Then
(a) f(x) is m.d. in \( (-\infty, 1) \)
(b) f(x) is m.i. in \( (1, +\infty) \)
(c) f(x) > 0 for x < 1
(d) f(x) < 0 for x > 1
Answer: (a) f(x) is m.d. in \( (-\infty, 1) \), (c) f(x) > 0 for x < 1

Question. Let \( f(x) = \frac{x^2 + 1}{[x]}, 1 \leq x \leq 3.9 \). [.] denote the greatest integer function. Then
(a) f(x) is m.d. in [1, 3.9]
(b) f(x) is m.i. in [1, 3.9]
(c) The greatest value of f(x) is \( \frac{1}{3} \times 16.21 \)
(d) The least value of f(x) is 2
Answer: (c) The greatest value of f(x) is \( \frac{1}{3} \times 16.21 \), (d) The least value of f(x) is 2

Question. Let \( f(x) = |x^2 - 3x - 4|, -1 \leq x \leq 4 \). Then
(a) f(x) is m.i. in [-1, 3/2)
(b) f(x) is m.d. in (3/2, 4]
(c) the maximum value of f(x) is 25/4
(d) the minimum value of f(x) is 0
Answer: (a) f(x) is m.i. in [-1, 3/2), (b) f(x) is m.d. in (3/2, 4], (c) the maximum value of f(x) is 25/4, (d) the minimum value of f(x) is 0

Question. Let \( f(x) = \phi(2 - x) + \phi(x) \) and \( \phi''(x) < 0 \) for \( x \in [0, 2] \). Then
(a) f(x) is m.i. in [0, 1]
(b) f(x) is m.d. in [0, 1]
(c) f(x) is m.i. in [1, 2]
(d) f(x) is m.d. in [1, 2]
Answer: (a) f(x) is m.i. in [0, 1], (d) f(x) is m.d. in [1, 2]

Question. Let \( f'(x) > 0 \) and \( g'(x) < 0 \) for all \( x \in R \). Then
(a) f{g(x)} > f{g(x + 1)}
(b) f{g(x)} > f{g(x - 1)}
(c) g{f(x)} > g{f(x + 1)}
(d) g{f(x)} > g{f(x - 1)}
Answer: (a) f{g(x)} > f{g(x + 1)}, (c) g{f(x)} > g{f(x + 1)}

Question. Let \( f(x) = 2x^2 - \log |x|, x \neq 0 \). Then f(x) is
(a) m.i. in \( (-\frac{1}{2}, 0) \cup (\frac{1}{2}, +\infty) \)
(b) m.d. in \( (-\frac{1}{2}, 0) \cup (\frac{1}{2}, +\infty) \)
(c) m.i. in \( (-\infty, \frac{1}{2}) \cup (0, \frac{1}{2}) \)
(d) m.d. in \( (-\infty, -\frac{1}{2}) \cup (0, \frac{1}{2}) \)
Answer: (a) m.i. in \( (-\frac{1}{2}, 0) \cup (\frac{1}{2}, +\infty) \), (d) m.d. in \( (-\infty, -\frac{1}{2}) \cup (0, \frac{1}{2}) \)

Question. Let \( f(x) = x^3 - 6x^2 + 15x + 3 \). Ten
(a) f(x) > 0 for all \( x \in R \)
(b) \( f(x) > f(x + 1) \) does not hold for any real x
(c) f(x) is invertible
(d) f(x) is a one-one function
Answer: (b) \( f(x) > f(x + 1) \) does not hold for any real x, (c) f(x) is invertible, (d) f(x) is a one-one function

Question. Let \( f(x) = a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x \), where \( a_i \)’s are real and f(x) = 0 has a positive root \( \alpha_0 \). Then
(a) \( f'(x) = 0 \) has a root \( \alpha_1 \) such that \( 0 < \alpha_1 < \alpha_0 \)
(b) \( f'(x) = 0 \) has at least one real root
(c) \( f''(x) = 0 \) has at least two real roots
(d) None of the options
Answer: (a) \( f'(x) = 0 \) has a root \( \alpha_1 \) such that \( 0 < \alpha_1 < \alpha_0 \), (b) \( f'(x) = 0 \) has at least one real root, (c) \( f''(x) = 0 \) has at least two real roots