JEE Mathematics Maxima and Minima MCQs Set 01

Question. If \( xy = a^2 \) and \( S = b^2x + c^2y \) where \( a, b \) and \( c \) are constants then the minimum value of \( S \) is
(a) \( abc \)
(b) \( bc\sqrt{a} \)
(c) \( 2abc \)
(d) None of the options
Answer: (c) 2abc

Question. The global minimum value of \( f(x) = x^4 – x^2 – 2x + 6 \) is
(a) 6
(b) 8
(c) 4
(d) nonexistent
Answer: (c) 4

Question. The maximum value of \( f(x) = 3 \cos^2 x + 4 \sin^2 x + \cos \frac{x}{2} + \sin \frac{x}{2} \) is
(a) 4
(b) \( 3 + \sqrt{2} \)
(c) \( 4 + \sqrt{2} \)
(d) None of the options
Answer: (c) \( 4 + \sqrt{2} \)

Question. If \( a > b > 0 \), the minimum value of \( a \sec \theta - b \tan \theta \) is
(a) \( b – a \)
(b) \( \sqrt{a^2 + b^2} \)
(c) \( \sqrt{a^2 - b^2} \)
(d) \( 2\sqrt{a^2 - b^2} \)
Answer: (c) \( \sqrt{a^2 - b^2} \)

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. The function \( f(x) = \frac{x}{1 + x \tan x} \) has
(a) one point of minimum in the interval \( (0, \pi/2) \)
(b) one point of maximum in the interval \( (0, \pi/2) \)
(c) no point of maximum, no point of minimum in the interval \( (0, \pi/2) \)
(d) two points of maxima in the interval \( (0, \pi/2) \)
Answer: (b) one point of maximum in the interval \( (0, \pi/2) \)

Question. Let \( f(x) = e^x \sin x \) be the equation of a curve. If at \( x = a \), \( 0 \leq a \leq 2\pi \), the slope of the tangent is the maximum then the value of \( a \) is
(a) \( \pi/2 \)
(b) \( 3\pi/2 \)
(c) \( \pi \)
(d) \( \pi/4 \)
Answer: (a) \( \pi/2 \)

Question. Let the tangent to the graph of \( y = f(x) \) at the point \( x = a \) be parallel to the x-axis, and let \( f'(a – h) > 0 \) and \( f'(a + h) < 0 \), where \( h \) is a very small positive number. Then the ordinate of the point is
(a) a maximum
(b) a minimum
(c) both a maximum and a minimum
(d) neither a maximum nor a minimum
Answer: (a) a maximum

Question. Let \( f(x) = x^3 + 3x^2 – 9x + 2 \). Then
(a) \( f(x) \) has a maximum at \( x = 1 \)
(b) \( f(x) \) has neither a minimum nor a maximum at \( x = -3 \)
(c) \( f(x) \) has a minimum at \( x = 1 \)
(d) None of the options
Answer: (c) \( f(x) \) has a minimum at \( x = 1 \)

Question. If \( f(x) = a \log_e |x| + bx^2 + x \) has extremums at \( x = 1 \) and \( x = 3 \) then
(a) \( a = -3/4, b = -1/8 \)
(b) \( a = 3/4, b = -1/8 \)
(c) \( a = -3/4, b = 1/8 \)
(d) None of the options
Answer: (a) \( a = -3/4, b = -1/8 \)

Question. The maximum value of \( (\frac{1}{x})^{2x^2} \) is
(a) \( e \)
(b) \( \sqrt{e} \)
(c) 1
(d) None of the options
Answer: (b) \( \sqrt{e} \)

Question. The maximum ordinate of a point on the graph of the function \( f(x) = \sin x(1 + \cos x) \) is
(a) \( \frac{2 + \sqrt{3}}{4} \)
(b) \( \frac{3\sqrt{3}}{4} \)
(c) 1
(d) None of the options
Answer: (b) \( \frac{3\sqrt{3}}{4} \)

Question. If \( \theta + \phi = \frac{\pi}{3} \) then \( \sin \theta \cdot \sin \phi \) has a maximum value at \( \theta = \)
(a) \( \frac{\pi}{6} \)
(b) \( \frac{2\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (a) \( \frac{\pi}{6} \)

Question. Let \( f(x) = x^3 – 6x^2 + 12x – 3 \). Then at \( x = 2, f(x) \) has
(a) a maximum
(b) a minimum
(c) both a maximum and a minimum
(d) neither a maximum nor a minimum
Answer: (d) neither a maximum nor a minimum

Question. Let \( f(x) = (x – p)^2 + (x – q)^2 + (x – r)^2 \). Then \( f(x) \) has a minimum at \( x = \lambda \), where \( \lambda \) is equal to
(a) \( \frac{p + q + r}{3} \)
(b) \( \sqrt[3]{pqr} \)
(c) \( \frac{3}{\frac{1}{p} + \frac{1}{q} + \frac{1}{r}} \)
(d) None of the options
Answer: (a) \( \frac{p + q + r}{3} \)

Question. Let \( f(x) = 1 + 2x^2 + 2^2x^4 + \dots + 2^{10}x^{20} \). Then \( f(x) \) has
(a) more than one minimum
(b) exactly one minimum
(c) at least one maximum
(d) None of the options
Answer: (b) exactly one minimum

Question. Let \( f(x) = \frac{a}{x} + x^2 \). If it has a maximum at \( x = -3 \) then \( a \) is
(a) -1
(b) 16
(c) 1
(d) None of the options
Answer: (d) None of the options

Question. Let \( f(x) \) be a function such that \( f'(a) \neq 0 \). Then at \( x = a, f(x) \)
(a) cannot have a maximum
(b) cannot have a minimum
(c) must have neither a maximum nor a minimum
(d) None of the options
Answer: (d) None of the options

Question. Let the function \( f(x) \) be defined as below.
\( f(x) = \sin^{-1} \lambda + x^2, 0 < x < 1 \)
\( 2x, x \geq 1 \)
\( f(x) \) can have a minimum at \( x = 1 \) if the value of \( \lambda \) is
(a) 1
(b) -1
(c) 0
(d) None of the options
Answer: (d) None of the options

Question. Let \( x \) be a number which exceeds its square by the greatest possible quantity. Then \( x \) is equal to
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{3}{4} \)
(d) None of the options
Answer: (a) \( \frac{1}{2} \)

Question. The sum of two nonzero numbers is 8. The minimum value of the sum of their reciprocals is
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{8} \)
(d) None of the options
Answer: (b) \( \frac{1}{2} \)

Question. In a \( \Delta ABC, \angle B = 90^\circ \) and \( b + a = 4 \). The area of the triangle is the maximum when \( \angle C \) is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{3} \)
(d) None of the options
Answer: (c) \( \frac{\pi}{3} \)

Question. The point (0, 3) is nearest to the curve \( x^2 = 2y \) at
(a) \( (2\sqrt{2}, 0) \)
(b) (0, 0)
(c) (2, 2)
(d) None of the options
Answer: (c) (2, 2)

Question. If \( \lambda, \mu \) be real numbers such that \( x^3 − \lambda x^2 + \mu x – 6 = 0 \) has its roots real and positive then the minimum value of \( \mu \) is
(a) \( 3 \times \sqrt[3]{36} \)
(b) 11
(c) 0
(d) None of the options
Answer: (a) \( 3 \times \sqrt[3]{36} \)

Question. Let \( f(x) = x + x^{-1} \). Then
(a) \( f(x) \) has a maximum but no minimum
(b) \( f(x) \) has no maximum but a minimum
(c) \( f(x) \) has a maximum and a minimum
(d) \( \text{max } f(x) < \text{min } f(x) \)
Answer: (c) \( f(x) \) has a maximum and a minimum, (d) \( \text{max } f(x) < \text{min } f(x) \)

Question. Let \( f(x) = ax^3 + bx^2 + cx + 1 \) have extrema at \( x = \alpha, \beta \) such that \( \alpha\beta < 0 \) and \( f(\alpha) \cdot f(\beta) < 0 \). Then the equation \( f(x) = 0 \) has
(a) three equal real roots
(b) three distinct real roots
(c) one positive root if \( f(\alpha) < 0 \) and \( f(\beta) > 0 \)
(d) one negative root if \( f(\alpha) > 0 \) and \( f(\beta) < 0 \)
Answer: (b) three distinct real roots, (c) one positive root if \( f(\alpha) < 0 \) and \( f(\beta) > 0 \), (d) one negative root if \( f(\alpha) > 0 \) and \( f(\beta) < 0 \)

Question. The critical point(s) of \( f(x) = \frac{|2 - x|}{x^2} \) is (are)
(a) x = 0
(b) x = 2
(c) x = 4
(d) None of the options
Answer: (a) x = 0, (b) x = 2, (c) x = 4

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) None of the options
Answer: (b) 1, (c) -1

Question. Let \( f(x) = x^3 + 3x^2 + 2x + 2 \). Then, at \( x = -1 \)
(a) \( f(x) \) has a maximum
(b) \( f(x) \) has a minimum
(c) \( f'(x) \) has a maximum
(d) \( f'(x) \) has a minimum
Answer: (d) \( f'(x) \) has a minimum

Question. The function \( f(x) = x^2 + \frac{\lambda}{x} \) has a
(a) minimum at x = 2 if \( \lambda = 16 \)
(b) maximum at x = 2 if \( \lambda = 16 \)
(c) maximum for no real value of \( \lambda \)
(d) point of inflection at x = 1 if \( \lambda = -1 \)
Answer: (a) minimum at x = 2 if \( \lambda = 16 \), (c) maximum for no real value of \( \lambda \), (d) point of inflection at x = 1 if \( \lambda = -1 \)

Question. Let \( f(x) = (x – 1)^4 \cdot (x – 2)^n, n \in N \). Then \( f(x) \) has
(a) a maximum at x = 1 if n is odd
(b) a maximum x = 1 if n is even
(c) a minimum at x = 2 if n is even
(d) a maximum at x = 2 if n is odd
Answer: (a) a maximum at x = 1 if n is odd, (c) a minimum at x = 2 if n is even