JEE Mathematics Maxima and Minima MCQs Set C

Question. If p and q are positive real numbers such that \( p^2 + q^2 = 1 \), then the maximum value of (p + q) is
(a) 2
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) \( \sqrt{2} \)
Answer: (d) \( \sqrt{2} \)

Question. The function \( f(x) = \frac{x}{2} + \frac{2}{x} \) has a local minimum at
(a) x = -2
(b) x = 0
(c) x = 1
(d) x = 2
Answer: (d) x = 2

Question. If x is real, the maximum value of \( \frac{3x^2 + 9x + 17}{3x^2 + 9x + 7} \) is
(a) 41
(b) 1
(c) 17/7
(d) 1/4
Answer: (a) 41

Question. A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is
(a) \( \sqrt{\frac{x^3}{8}} \)
(b) \( \frac{1}{2} x^2 \)
(c) \( \pi x^2 \)
(d) \( \frac{3}{2} x^2 \)
Answer: (b) \( \frac{1}{2} x^2 \)

Question. If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2 x + 1 \), where a > 0, attains its maximum and minimum at p and q respectively such that \( p^2 = q \), then a equals
(a) 3
(b) 1
(c) 2
(d) 1/2
Answer: (c) 2

Question. The maximum value \( x^3 - 3x \) in the interval [0, 2] is
(a) 1
(b) 2
(c) 0
(d) -2
Answer: (b) 2

Question. Minimum value of \( \frac{1}{3 \sin\theta - 4 \cos\theta + 7} \) is
(a) 7/12
(b) 5/12
(c) 1/12
(d) 1/6
Answer: (c) 1/12

Question. The minimum value of \( (x - p)^2 + (x - q)^2 + (x - r)^2 \) will be at x equals to
(a) pqr
(b) \( \sqrt[3]{pqr} \)
(c) \( \frac{p + q + r}{3} \)
(d) \( p^2 + q^2 + r^2 \)
Answer: (c) \( \frac{p + q + r}{3} \)

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

Question. The co-ordinate of the point for minimum value of z = 7x - 8y subject to the conditions \( x + y - 20 \leq 0 \), \( y \geq 5 \), \( x \geq 0 \), \( y \geq 0 \)
(a) (20, 0)
(b) (15, 5)
(c) (0, 5)
(d) (0, 20)
Answer: (d) (0, 20)

Question. The maximum value of \( \cos \alpha_1 \cdot \cos \alpha_2 \cdot \cos \alpha_3 \dots \cos \alpha_n \) under the restriction \( 0 \leq \alpha_1, \alpha_2, \dots , \alpha_n \leq \frac{\pi}{2} \) and \( \cot \alpha_1 \cdot \cot \alpha_2 \dots \cot \alpha_n = 1 \) is
(a) \( 1/2^{n/2} \)
(b) \( 1/2^n \)
(c) \( -1/2^n \)
(d) 1
Answer: (a) \( 1/2^{n/2} \)

Question. The point on the curve \( 4x^2 + a^2 y^2 = 4a^2 \), \( 4 < a^2 < 8 \), that is farthest from the point (0, -2) is
(a) (2, 0)
(b) (0, 2)
(c) (2, -2)
(d) (-2, 2)
Answer: (b) (0, 2)

Question. The equation \( x^3 - 3x + [a] = 0 \), will have three real and distinct roots if (where [*] denotes the greatest integer function)
(a) \( a \in (-\infty, 2) \)
(b) \( a \in (0, 2) \)
(c) \( a \in (-\infty, 2) \cup (0, \infty) \)
(d) \( a \in [-1, 2) \)
Answer: (d) \( a \in [-1, 2) \)

Question. Let \( f(x) = \sin \frac{\{x\}}{a} + \cos \frac{\{x\}}{a} \). Then the set of values of a for which f can attain its maximum values is (where a>0 and { * } denotes the fractional part function)
(a) \( \left(0, \frac{4}{\pi}\right) \)
(b) \( \left(\frac{4}{\pi}, \infty\right) \)
(c) \( (0, \infty) \)
(d) None of the options
Answer: (a) \( \left(0, \frac{4}{\pi}\right) \)

Question. A possible ordered pair (a, b) such that all the local extremum values of the function \( f(x) = x^3 + ax^2 - 9x + b \) are positive and the local minimum value occurs at point x = 1 is
(a) (3, 5)
(b) (3, 6)
(c) (3, 4)
(d) (3, 3)
Answer: (b) (3, 6)

Question. A function is defined as \( f(x) = ax^2 - b|x| \) where a and b are constants then at x = 0 we will have a maxima of f(x) if
(a) a > 0, b > 0
(b) a > 0, b < 0
(c) a < 0, b < 0
(d) a < 0, b > 0
Answer: (a) a > 0, b > 0

Question. A and B are the points (2, 0) and (0, 2) respectively. The coordinates of the point P on the line \( 2x + 3y + 1 = 0 \) are
(a) (7, -5) if |PA - PB| is maximum
(b) \( \left(\frac{1}{5}, \frac{1}{5}\right) \) if |PA - PB| is maximum
(c) (7, -5) if |PA - PB| is minimum
(d) \( \left(\frac{1}{5}, \frac{1}{5}\right) \) if |PA - PB| is minimum
Answer: (a) (7, -5) if |PA - PB| is maximum

Question. The maximum value of \( f(x) = 2bx^2 - x^4 - 3b \) is g(b), where b > 0, if b varies then the minimum value of g(b) is
(a) 3/2
(b) 9/2
(c) -9/4
(d) -9/2
Answer: (c) -9/4

Question. Number of solution(s) satisfying the equation, \( 3x^2 - 2x^3 = \log_2 (x^2 + 1) - \log_2 x \) is
(a) 1
(b) 2
(c) 3
(d) None of the options
Answer: (a) 1

Question. If \( a^2 x^4 + b^2 y^4 = c^6 \), then the maximum value of xy is
(a) \( \frac{c^3}{2ab} \)
(b) \( \frac{c^3}{\sqrt{2|ab|}} \)
(c) \( \frac{c^3}{ab} \)
(d) \( \frac{c^3}{\sqrt{|ab|}} \)
Answer: (b) \( \frac{c^3}{\sqrt{2|ab|}} \)

Question. Maximum and minimum value of \( f(x) = \max (\sin t) \), \( 0 < t < x \), \( 0 \leq x \leq 2\pi \) are
(a) 1, 0
(b) 1, -1
(c) 0, -1
(d) None of the options
Answer: (a) 1, 0

Question. The greatest value of \( f(x) = (x + 1)^{1/3} - (x - 1)^{1/3} \) in [0, 1] is
(a) 1
(b) 2
(c) 3
(d) \( 2^{1/3} \)
Answer: (b) 2

Question. The function 'f' is defined by \( f(x) = x^p (1 - x)^q \) for all \( x \in \mathbb{R} \), where p, q are positive integers, has a maximum value, for x equal to
(a) \( \frac{pq}{p + q} \)
(b) 1
(c) 0
(d) \( \frac{p}{p + q} \)
Answer: (d) \( \frac{p}{p + q} \)

Question. The maximum slope of the curve \( y = -x^3 + 3x^2 + 2x - 27 \) will be
(a) -165/8
(b) -27
(c) 5
(d) None of the options
Answer: (c) 5

Advanced Subjective Questions

Question. A trapezium ABCD is inscribed into a semicircle of radius \(l\) so that the base AD of the trapezium is a diameter and the vertices B & C lie on the circumference. Find the base angle \(\theta\) of the trapezium ABCD which has the greatest perimeter.
Answer: \(CD = 2l \cos \theta\)
\(CM = 2l \cos^2 \theta\)
\(P = 4l(1 + \cos \theta - \cos^2 \theta)\)
\(\frac{dp}{d\theta} = 0 \Rightarrow \theta = \frac{\pi}{3}, 0\)
\(\frac{d^2p}{d\theta^2} \bigg|_{\pi/3} < 0 \text{ so } \theta = \frac{\pi}{3}\)

Question. If \(y = \frac{ax + b}{(x - 1)(x - 4)}\) has a turning value at \((2, -1)\) find a & b and show that the turning value is a maximum.
Answer: \(\frac{dy}{dx} = \frac{(2x - 5)(ax + b) - a(x^2 - 5x + 4)}{[(x - 1)(x - 4)]^2}\)
at \(x = 2\) & \(-1\)
\(\frac{dy}{dx} = 0 \Rightarrow a = 1\) & \(b = 0\)
so Now \(y = \frac{x}{x^2 - 5x + 4}\)
Now find \(\frac{dy}{dx}\) & check maximum value.

Question. Prove that among all triangles with a given perimeter, the equilateral triangle has the maximum area.
Answer: Let a, b, c be the sides and \(a + b + c = 2s\)
\(\Delta^2 = s(s - a)(s - b)(s - c)\)
by \(AM \ge GM\)
\(\Rightarrow \frac{(s - a) + (s - b) + (s - c)}{3} \ge \sqrt[3]{(s - a)(s - b)(s - c)}\)
\(\therefore \frac{3s - (a + b + c)}{3} \ge \sqrt[3]{(s - a)(s - b)(s - c)}\)
or \(\left(\frac{s}{3}\right)^3 \ge (s - a)(s - b)(s - c)\)
\(\therefore \frac{s^4}{27} \ge s(s - a)(s - b)(s - c)\)
\(\therefore \frac{s^4}{27} \ge \Delta^2 \Rightarrow \Delta \le \frac{s^2}{3\sqrt{3}}\)
\(\therefore \Delta\) has the maximum value \(\frac{s^2}{3\sqrt{3}}\) and it takes place when \(AM = GM\). hence \(a = b = c\)

Question. A sheet of poster has its area 18 m\(^2\). The margin at the top & bottom are 75 cms and at the sides 50 cms. What are the dimensions of the poster if the area of the printed space is maximum ?
Answer: \(xy = 18\) ;
after marging ; \(l = x - 3/4\), \(w = y - 1/2\)
so \(A' = xy - \frac{x}{2} - \frac{3}{4}y + \frac{3}{8}\)
\(A' = 18 - \frac{x}{2} - \frac{3}{4} \cdot \frac{18}{x}\)
\(\frac{dA'}{dx} = 0 \Rightarrow x = 3\sqrt{3}\)
\(y = 2\sqrt{3}\)

Question. The mass of a cell culture at time t is given by, \(M(t) = \frac{3}{1 + 4e^{-t}}\)
(a) Find \(\lim_{t \to -\infty} M(t)\) and \(\lim_{t \to \infty} M(t)\)
(b) Show that \(\frac{dM}{dt} = \frac{1}{3} M(3 - M)\)
(c) Find the maximum rate of growth of M and also the value of t at which occurs.
Answer: \(M(t) = \frac{3}{1 + 4e^{-t}}\)
\(\lim_{t \to -\infty} M(t) = 0\) & \(\lim_{t \to \infty} M(t) = 3\)
\(M = 3 - \frac{12}{e^t + 4}\)
\(\frac{dM}{dt} = \frac{-12e^t}{(e^t + 4)^2}\)
put in the given relation.

Question. Depending on the values of p \(\in \mathbb{R}\), find the value of 'a' for which the equation \(x^3 + 2 px^2 + p = a\) has three distinct real roots.
Answer: Let \(f(x) = x^3 + 2Px^2 + P - a\)
\(f'(x) = 3x^2 + 4Px = 0\)
\(x = 0, x = -\frac{4P}{3}\)
\(f(0) \cdot f\left(-\frac{4P}{3}\right) < 0\)
\((P - a) \left[-\frac{64}{27}P^3 + \frac{32}{9}P^3 + P - a\right] < 0\)
\((P - a) \left[\frac{32}{27}P^3 + P - a\right] < 0\)
\((a - P) \left[a - P - \frac{32}{27}P^3\right] < 0\)
If \(P > 0\)
\(P < a < P + \frac{32}{27}P^3\)
If \(P < 0\)
\(\frac{32}{27}P^3 + P < a < P\)