JEE Mathematics Maxima and Minima MCQs Set 02

Question. \( f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + \dots + 100x^{100} \) is polynomial in a real variable x, then f(x) has
(a) neither a maximum nor a minimum
(b) only one maximum
(c) only one minimum
(d) one maximum and one minimum
Answer: (c) only one minimum

Question. On the interval [0, 1] the function \( x^{25}(1 - x)^{75} \) takes its maximum value at
(a) 0
(b) 1/2
(c) 1
(d) 1/4
Answer: (d) 1/4

Question. The product of minimum value of \( x^x \) and maximum value of \( \left(\frac{1}{x}\right)^x \) is
(a) e
(b) \( e^{-1} \)
(c) 1
(d) \( e^2 \)
Answer: (c) 1

Question. The minimum value of the function defined by f(x) = max (x, x + 1, 2 - x) is
(a) 0
(b) 1/2
(c) 1
(d) 3/2
Answer: (d) 3/2

Question. Let \( f(x) = \begin{cases} \sin \frac{\pi}{2}x, & 0 \leq x < 1 \\ 3 - 2x, & x \geq 1 \end{cases} \) then
(a) f(x) has local maxima at x = 1
(b) f(x) has local minima at x = 1
(c) f(x) does not have any local extrema at x = 1
(d) f(x) has a global minima at x = 1
Answer: (a) f(x) has local maxima at x = 1

Question. The greatest and the least values of the function, \( f(x) = 2 - \sqrt{1 + 2x + x^2} \), \( x \in [-2, 1] \) are
(a) 2, 1
(b) 2, -1
(c) 2, 0
(d) None of the options
Answer: (c) 2, 0

Question. Let f(x) = {x}, For f(x), x = 5 is (where {*} denotes the fractional part)
(a) a point of local maxima
(b) a point of local minima
(c) neither a point of local minima nor maxima
(d) None of the options
Answer: (b) a point of local minima

Question. The critical points of \( f(x) = \frac{|x - 1|}{x^2} \) lies at
(a) \( x \in \{1, 2\} \)
(b) \( x \in \{0, 1\} \)
(c) \( x \in \{2\} \)
(d) None of the options
Answer: (a) \( x \in \{1, 2\} \)

Question. The difference between the greatest and least values of the function \( f(x) = \sin 2x - x \) on \( [-\pi/2, \pi/2] \) is
(a) \( \frac{\sqrt{3} + \sqrt{2}}{2} \)
(b) \( \frac{\sqrt{3} + \sqrt{2}}{2} + \frac{\pi}{6} \)
(c) \( \frac{\pi}{2} \)
(d) \( \pi \)
Answer: (d) \( \pi \)

Question. The radius of a right circular cylinder of greatest curved surface which can be inscribed in a given right circular cone is
(a) one third that of the cone
(b) \( 1/\sqrt{2} \) times that of the cone
(c) 2/3 that of the cone
(d) 1/2 that of the cone
Answer: (d) 1/2 that of the cone

Question. The dimensions of the rectangle of maximum area that can be inscribed in the ellipse \( (x/4)^2 + (y/3)^2 = 1 \) are
(a) \( \sqrt{8}, \sqrt{2} \)
(b) 4, 3
(c) \( 2\sqrt{8}, 3\sqrt{2} \)
(d) \( \sqrt{2}, \sqrt{6} \)
Answer: (c) \( 2\sqrt{8}, 3\sqrt{2} \)

Question. The largest area of a rectangle which has one side on the x-axis and the two vertices on the curve \( y = e^{-x^2} \) is
(a) \( \sqrt{2} e^{-1/2} \)
(b) \( 2 e^{-1/2} \)
(c) \( e^{-1/2} \)
(d) None of the options
Answer: (a) \( \sqrt{2} e^{-1/2} \)

Question. The co-ordinates of the point on the curve \( x^2 = 4y \), which is at least distance from the line \( y = x - 4 \) is
(a) (2, 1)
(b) (-2, 1)
(c) (1, -2)
(d) (1, 2)
Answer: (a) (2, 1)

Question. \( f(x) = \begin{cases} \tan^{-1}x, & |x| < \frac{\pi}{2} \\ \frac{\pi}{2} - |x|, & |x| \geq \frac{\pi}{2} \end{cases} \) then
(a) f(x) has no point of local maxima
(b) f(x) has only one point of local maxima
(c) f(x) has exactly two points of local maxima
(d) f(x) has exactly two points of local minima
Answer: (b) f(x) has only one point of local maxima

Question. Let \( f(x) = \begin{cases} x^3 - x^2 + 10x - 5, & x \leq 1 \\ -2x + \log_2 (b^2 - 2), & x > 1 \end{cases} \) the set of values of b for which f(x) has greatest value at x = 1 is given by
(a) \( 1 \leq b \leq 2 \)
(b) \( b = \{1, 2\} \)
(c) \( b \in (-\infty, -1) \)
(d) \( [-\sqrt{130}, -\sqrt{2}) \cup (\sqrt{2}, \sqrt{130}] \)
Answer: (d) \( [-\sqrt{130}, -\sqrt{2}) \cup (\sqrt{2}, \sqrt{130}] \)

Question. The set of values of p for which the extrema of the function, \( f(x) = x^3 - 3px^2 + 3(p^2 - 1) x + 1 \) lie in the interval (-2, 4) is
(a) (-3, 5)
(b) (-3, 3)
(c) (-1, 3)
(d) (-1, 4)
Answer: (c) (-1, 3)

Question. Four points A, B, C, D lie in that order on the parabola \( y = ax^2 + bx + c \). The co-ordinates of A, B & D are known as A(-2, 3); B(-1, 1) and D(2, 7). The co-ordinates of C for which the area of the quadrilateral ABCD is greatest is
(a) (1/2, 7/4)
(b) (1/2, -7/4)
(c) (-1/2, 7/4)
(d) None of the options
Answer: (a) (1/2, 7/4)

Question. In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is \( l \). The altitude of the prism for which the volume is greatest is
(a) \( \frac{l}{2} \)
(b) \( \frac{l}{\sqrt{3}} \)
(c) \( \frac{l}{3} \)
(d) \( \frac{l}{4} \)
Answer: (b) \( \frac{l}{\sqrt{3}} \)

Question. Two vertices of a rectangle are on the positive x-axis. The other two vertices lie on the lines \( y = 4x \) and \( y = -5x + 6 \). Then the maximum area of the rectangle is
(a) 4/3
(b) 3/5
(c) 4/5
(d) 3/4
Answer: (c) 4/5

Question. A variable point P is chosen on the straight line \( x + y = 4 \) and tangents PA and PB are drawn from it to circle \( x^2 + y^2 = 1 \). Then the position of P for the smallest length of chord of contact AB is
(a) (3, 1)
(b) (0, 4)
(c) (2, 2)
(d) (4, 0)
Answer: (c) (2, 2)

Question. The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The fraction of width folded over if the area of the folded part is minimum is
(a) 5/8
(b) 2/3
(c) 3/4
(d) 4/5
Answer: (b) 2/3

Question. If \( x_1 \) and \( x_2 \) are abscissa of two points on the curve \( f(x) = x - x^2 \) in the interval [0, 1], then maximum value of the expression \( (x_1 + x_2) - (x_1^2 + x_2^2) \) is
(a) 1/2
(b) 1/4
(c) 1
(d) 2
Answer: (a) 1/2

Question. The maximum area of the rectangle whose sides pass through the angular points of a given rectangle of sides a and b is
(a) 2 (ab)
(b) \( \frac{1}{2} (a + b)^2 \)
(c) \( \frac{1}{2} (a^2 + b^2) \)
(d) None of the options
Answer: (b) \( \frac{1}{2} (a + b)^2 \)

Question. Least value of the function, \( f(x) = 2x^2 - 1 + \frac{2}{2x^2 + 1} \) is
(a) 0
(b) 3/2
(c) 2/3
(d) 1
Answer: (d) 1

Advanced Subjective Questions

Question. A cubic \(f(x)\) vanishes at \(x = -2\) & has relative minimum/maximum at \(x = -1\) and \(x = 1/3\). If \(\int_{-1}^{1} f(x)dx = \frac{14}{3}\), find the cubic \(f(x)\).
Answer: \(f'(x)\) is a quadratic & \(f'(-1) = 0\) and \(f'\left(\frac{1}{3}\right) = 0\)
so Let \(f'(x) = a(x + 1)\left(x - \frac{1}{3}\right)\)
\(\Rightarrow f(x) = a\left(\frac{x^3}{3} + \frac{x^2}{3} - \frac{x}{3}\right) + b\)
(\(b = 2\) at \(x = -2\))
\(y = 0\)
\(\frac{a}{3}\int_{-1}^{1} \left(x^3 + x^2 - x + 2\right) dx = \frac{14}{3}\)
\(\Rightarrow a = 3\) so \(f(x) = x^3 + x^2 - x + 2\)

Question. Find the greatest & least value for the function ;
(a) \(y = x + \sin 2x\), \(0 \le x \le 2\pi\)
(b) \(y = 2 \cos 2x - \cos 4x\), \(0 \le x \le \pi\)
Answer: (a) \(y = x + \sin 2x\)
\(\frac{dy}{dx} = 1 + 2 \cos 2x\)
\(\frac{dy}{dx} = 0 \Rightarrow \cos 2x = -\frac{1}{2}\)
\(x = \frac{\pi}{3}, \frac{2\pi}{3}\)
\(f(0) = 0\)
\(f(2\pi) = 2\pi\)
\(f\left(\frac{\pi}{3}\right) = \frac{\pi}{3} + \frac{\sqrt{3}}{2}\)
\(f\left(\frac{2\pi}{3}\right) = \frac{2\pi}{3} - \frac{\sqrt{3}}{2}\)
max. value = \(2\pi\)
min. value = \(0\)
(b) \(y = 2 \cos 2x - \cos 4x\)
\(\frac{dy}{dx} = -4 \sin 2x + 4 \sin 4x = 0\)
\(\Rightarrow x = 0, \frac{\pi}{2}, \frac{\pi}{6}\)
\(f(0) = 1\)
\(f(\pi) = 1\)
\(f\left(\frac{\pi}{2}\right) = -3\)
\(f\left(\frac{\pi}{6}\right) = \frac{3}{2}\)
min. value = \(-3\)
max. value = \(\frac{3}{2}\)

Question. Suppose \(f(x)\) is real valued polynomial function of degree 6 satisfying the following conditions ;
(a) \(f\) has minimum value at \(x = 0\) and 2
(b) \(f\) has maximum value at \(x = 1\)
(c) for all \(x\), \(\lim_{x \to 0} \frac{1}{x} \ln \begin{vmatrix} \frac{f(x)}{x} & 1 & 0 \\ \frac{1}{x} & 1 & 1 \\ 1 & 0 & \frac{1}{x} \end{vmatrix} = 2\). Determine \(f(x)\).
Answer: The value of the determinant is
\(D = \frac{f(x)}{x^3} + 1\)
\(\lim_{x \to 0} \frac{1}{x} \ln \left(\frac{f(x)}{x^3} + 1\right) = \lim_{x \to 0} \ln \left(\frac{f(x)}{x^3} + 1\right)^{1/x} = 2\) (given)
For the existence of limit coefficient of \(x^3\), \(x^2\), \(x\) & constant term of \(f(x)\) is zero.
Now \(\lim_{x \to 0} \ln e^{\frac{f(x)}{x^3} \cdot \frac{1}{x}} = \lim_{x \to 0} \frac{ax^6 + bx^5 + cx^4}{x^4} = 2\)
\(\Rightarrow c = 2\)
\(f'(x) = x^3 (6ax^2 + 5bx + 8)\)
\(f'(1) = 0\) & \(f'(2) = 0 \Rightarrow 6a + 5b + 8 = 0\)
& \(24a + 10b + 8 = 0\)
on solving \(a = \frac{2}{3}\), \(b = -\frac{12}{5}\)
so \(f(x) = \frac{2}{3}x^6 - \frac{12}{5}x^5 + 2x^4\)

Question. Of all the lines tangent to the graph of the curve \(y = \frac{6}{x^2 + 3}\), find the equations of the tangent lines of minimum and maximum slope.
Answer: \(\frac{dy}{dx} = -\frac{12x}{(x^2 + 3)^2} = g(x)\) (let)
\(g'(x) = 0 \text{ at } x = \pm 1\)
maximum slope
\(g(-1) = 3/4\) & point \((-1, 3/2)\)
minimum slope
\(g(1) = - 3/4\) & point \((1, 3/2)\).

Question. A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost of the material per square foot for the bottom is 15 paise, for the top 25 paise and for the sides 20 paise. The labour charges for making the box are Rs. 3/-. Find the dimensions of the box when the cost is minimum.
Answer: \(x^2 h = 1000\)
Top portion = \(x^2\)
Base = \(x^2\)
Sides = \(4xh\)
\(E = 15 x^2 + 25 x^2 + 20 (4xh) + 300\)
\(E = 40 x^2 + 80 x \left(\frac{1000}{x^2}\right) + 300\)
\(E = 40 x^2 + \frac{80000}{x} + 300\)
\(\frac{dE}{dx} = 0 \Rightarrow x = 10\)
\(\frac{d^2E}{dx^2} = 80 + \frac{160000}{x^3} \bigg|_{x=10} > 0\)
\(\Rightarrow x = 10 \Rightarrow h = 10\)

Question. Find the area of the largest rectangle with lower base on the x-axis & upper vertices on the curve \(y = 12 - x^2\).
Answer: \(A = 2\alpha (12 - \alpha^2)\)
\(\frac{dA}{d\alpha} = 2 (12 - 3\alpha^2) = 0\)
\(\Rightarrow \alpha = \pm 2\)
\(\frac{d^2A}{d\alpha^2} = -12\alpha \big|_{\alpha=2} < 0\)
\(A_{\text{max}} = 4 (12 - 4) = 32\) sq. units