Class 11 Mathematics Limits And Derivatives MCQs Set 07

Question. The function \( f(x) = x^{1/3}(x - 1) \)
(a) has two inflection points
(b) has one point extremum
(c) is non-differentiable at \( x = 0 \)
(d) Range of \( f(x) \) is \( [-3 \times 2^{-8/3}, \infty) \)
Answer: (a), (b), (c), and (d) are all correct.

Question. The value of \( x \) for which the function \( f(x) = \int_0^x (1 - t^2) e^{-t^2/2} dt \) has an extremum is
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (b) 1 and (c) -1

Question. A tangent to the curve \( y = \int_0^x |t| dt \), which is parallel to the line \( y = x \), cuts off an intercept from the \( y \)-axis equals to
(a) 1
(b) -1/2
(c) 1/2
(d) -1
Answer: (b) -1/2

Question. If \( f(x) \) and \( g(x) \) are two positive and increasing functions, then
(a) \( (f(x))^{g(x)} \) is always increasing
(b) If \( (f(x))^{g(x)} \) is decreasing then \( f(x) < 1 \)
(c) if \( (f(x))^{g(x)} \) is increasing then \( f(x) > 1 \)
(d) If \( f(x) > 1 \), then \( (f(x))^{g(x)} \) is increasing
Answer: (b) If \( (f(x))^{g(x)} \) is decreasing then \( f(x) < 1 \) and (d) If \( f(x) > 1 \), then \( (f(x))^{g(x)} \) is increasing

Question. \( f(x) = \begin{cases} |x + 1|; & -2 < x < 0 \\ 2; & x = 0 \\ \sqrt[3]{1 - x}; & 0 < x < 1 \\ \sqrt{x + 1} & x \geq 1 \end{cases} \). Then \( f(x) \)
(a) has neither maximum nor minimum at \( x = 0 \)
(b) has maximum at \( x = 0 \)
(c) has neither maximum nor minimum at \( x = 1 \)
(d) no global maximum
Answer: (b) has maximum at \( x = 0 \) and (d) no global maximum

Question. If \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), for every real number \( x \), then the minimum value of \( f \)
(a) does not exist because \( f \) is unbounded
(b) is not attained even though \( f \) is bounded
(c) is equal to 1
(d) is equal to -1
Answer: (d) is equal to -1

Question. The number of values of \( x \) where the function \( f(x) = \cos x + \cos(\sqrt{2}x) \) attains its maximum is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (b) 1

Question. \( f(x) \) is cubic polynomial with \( f(2) = 18 \) and \( f(1) = -1 \). Also \( f(x) \) has local maxima at \( x = -1 \) and \( f'(x) \) has local minima at \( x = 0 \), then 
(a) the distance between \( (-1, 2) \), and \( (a, f(a)) \), where \( x = a \) is the point of local minima is \( 2\sqrt{5} \)
(b) \( f(x) \) is increasing for \( x \in [1, 2\sqrt{5}] \)
(c) \( f(x) \) has local minima at \( x = 1 \)
(d) the value of \( f(0) = 15 \)
Answer: (c) \( f(x) \) has local minima at \( x = 1 \)

Question. Let \( f(x) = \begin{cases} e^x, & 0 \leq x \leq 1 \\ 2 - e^{x-1}, & 1 < x \leq 2 \\ x - e, & 2 < x \leq 3 \end{cases} \), \( g(x) = \int_0^x f(t) dt, x \in [1,3] \) then \( g(x) \) has
(a) local maxima at \( x = 1 + \ln 2 \) and local minima at \( x = e \)
(b) local maxima at \( x = 1 \) and local minima at \( x = 2 \)
(c) no local maxima
(d) no local minima
Answer: (a) local maxima at \( x = 1 + \ln 2 \) and local minima at \( x = e \)

Question. For function \( f(x) = x \cos \frac{1}{x}, x \geq 1 \),
(a) for at least one \( x \) in interval \( [1, \infty), f(x + 2) - f(x) < 2 \)
(b) \( \lim_{x \to 0} f'(x) = 1 \)
(c) \( [1, \infty), f(x + 2) - f(x) > 2 \)
(d) \( f'(x) \) is strictly decreasing in the interval \( [1, \infty) \)
Answer: (a) for at least one \( x \) in interval \( [1, \infty), f(x + 2) - f(x) < 2 \) and (d) \( f'(x) \) is strictly decreasing in the interval \( [1, \infty) \)

Question. If \( f(x) = \int_0^x e^{t^2} (t - 2)(t - 3) dt \) for all \( x \in (0, \infty) \), then
(a) \( f \) has a local maximum at \( x = 2 \)
(b) \( f \) is decreasing on \( (2, 3) \)
(c) there exists some \( c \in (0, \infty) \) such that \( f''(c) = 0 \)
(d) \( f \) has a local minimum at \( x = 3 \)
Answer: (a), (b), (c), and (d) are all correct.

ASSERTION & REASON QUESTIONS

Question. Statement-1: Both \( \sin x \) and \( \cos x \) are decreasing functions in the interval \( \left( \frac{\pi}{2}, \pi \right) \)
Statement-2: If a differentiable function decreases in an interval \( (a, b) \), then its derivative also decreasing in \( (a, b) \). Which of the following is true?
(a) both st1 and st2 are wrong
(b) both st1 and st2 are correct but st2 is not correct explanation for st1
(c) both st1 and st2 are correct and st2 is correct explanation for st1
(d) st1 is correct and st2 is wrong
Answer: (d) st1 is correct and st2 is wrong