JEE Mathematics Maxima and Minima MCQs Set D

Question. The least area of a circle circumscribing any right triangle of area S is
(a) \( \pi S \)
(b) \( 2 \pi S \)
(c) \( \sqrt{2} \pi S \)
(d) \( 4 \pi S \)
Answer: (a) \( \pi S \)

Question. Two points A(1, 4) & B(3, 0) are given on the ellipse \( 2x^2 + y^2 = 18 \). The co-ordinates of a point C on the ellipse such that the area of the triangle ABC is greatest is
(a) \( (\sqrt{6}, \sqrt{6}) \)
(b) \( (-\sqrt{6}, \sqrt{6}) \)
(c) \( (\sqrt{6}, -\sqrt{6}) \)
(d) \( (-\sqrt{6}, -\sqrt{6}) \)
Answer: (d) \( (-\sqrt{6}, -\sqrt{6}) \)

Question. The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height must be equal to
(a) 1/3
(b) 2/3
(c) \( 1/\sqrt{3} \)
(d) 1
Answer: (c) \( 1/\sqrt{3} \)

Question. Let \( f(x) = 5x - 2x^2 + 2 \); \( x \in \mathbb{N} \) then the maximum value of f(x) is
(a) 8
(b) 5
(c) 4
(d) 41/8
Answer: (b) 5

Question. The maximum value of f(x), if \( f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} \), x \(\in\) domain of f
(a) -1
(b) 2
(c) 1
(d) 1/2
Answer: (d) 1/2

Question. If \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), for every real number, then minimum value of \( f(x) \)
(a) does not exist
(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. Let \( f(x) = \frac{40}{3x^4 + 8x^3 - 18x^2 + 60} \), consider the following statement about \( f(x) \).
(a) \( f(x) \) has local minima at \( x = 0 \)
(b) \( f(x) \) has local maxima at \( x = 0 \)
(c) absolute maximum value of \( f(x) \) is not defined
(d) \( f(x) \) is local maxima at \( x = -3, x = 1 \)
Answer: (a) \( f(x) \) has local minima at \( x = 0 \), (c) absolute maximum value of \( f(x) \) is not defined, (d) \( f(x) \) is local maxima at \( x = -3, x = 1 \)

Question. If \( f(x) = a \ln |x| + bx^2 + x \) has its extremum values at \( x = -1 \) and \( x = 2 \), then
(a) \( a = 2, b = -1 \)
(b) \( a = 2, b = -1/2 \)
(c) \( a = - 2, b = 1/2 \)
(d) None of the options
Answer: (b) \( a = 2, b = -1/2 \)

Question. Let \( f(x) = (x^2 - 1)^n (x^2 + x + 1) \) then \( f(x) \) has local extremum at \( x = 1 \) when
(a) \( n = 2 \)
(b) \( n = 3 \)
(c) \( n = 4 \)
(d) \( n = 6 \)
Answer: (a) \( n = 2 \), (c) \( n = 4 \), (d) \( n = 6 \)

Question. An extremum value of the function \( f(x) = (\arcsin x)^3 + (\arccos x)^3 \) is
(a) \( \frac{7\pi^3}{8} \)
(b) \( \frac{\pi^3}{8} \)
(c) \( \frac{\pi^3}{32} \)
(d) \( \frac{\pi^3}{16} \)
Answer: (a) \( \frac{7\pi^3}{8} \), (c) \( \frac{\pi^3}{32} \)

Question. If \( f(x) = \frac{x}{1 + x \tan x} \), \( x \in \left(0, \frac{\pi}{2}\right) \), then
(a) \( f(x) \) has exactly one point of minima
(b) \( f(x) \) has exactly one point of maxima
(c) \( f(x) \) is increasing in \( \left(0, \frac{\pi}{2}\right) \)
(d) maxima occurs at \( x_0 \) where \( x_0 = \cos x_0 \)
Answer: (b) \( f(x) \) has exactly one point of maxima, (d) maxima occurs at \( x_0 \) where \( x_0 = \cos x_0 \)

Question. If \( f(x) = \begin{cases} -\sqrt{1 - x^2}, & 0 \le x \le 1 \\ -x, & x > 1 \end{cases} \), then
(a) Maximum of \( f(x) \) exist at \( x = 1 \)
(b) Maximum of \( f(x) \) doesn't exists
(c) Minimum of \( f^{-1}(x) \) exist at \( x = -1 \)
(d) Minimum of \( f^{-1}(x) \) exist at \( x = 1 \)
Answer: (a) Maximum of \( f(x) \) exist at \( x = 1 \), (c) Minimum of \( f^{-1}(x) \) exist at \( x = -1 \)

Question. If the function \( y = f(x) \) is represented as, \( x = \phi(t) = t^3 - 5t^2 - 20t + 7 \), \( y = \psi(t) = 4t^3 + 4t^2 - 18t + 3 \ ( | t | < 2) \), then
(a) \( y_{\text{max}} = 12 \)
(b) \( y_{\text{max}} = 14 \)
(c) \( y_{\text{min}} = -67/4 \)
(d) \( y_{\text{min}} = -69/4 \)
Answer: (b) \( y_{\text{max}} = 14 \), (d) \( y_{\text{min}} = -69/4 \)

Question. For the function \( f(x) = x^{2/3} \), which of the following statement(s) is/are true ?
(a) \( \frac{dy}{dx} \) at the origin is non existent
(b) equation of the tangent at the origin is \( x = 0 \)
(c) \( f(x) \) has an extremum at \( x = 0 \)
(d) origin is the point of inflection
Answer: (a) \( \frac{dy}{dx} \) at the origin is non existent, (c) \( f(x) \) has an extremum at \( x = 0 \)

Question. If \( \lim_{x \to a} f(x) = \lim_{x \to a} [f(x)] \) and \( f(x) \) is non-constant continuous function, then (where [ * ] denotes the greatest integer function)
(a) \( \lim_{x \to a} f(x) \) is integer
(b) \( \lim_{x \to a} f(x) \) is non-integer
(c) \( f(x) \) has local maximum at \( x = a \)
(d) \( f(x) \) has local minima at \( x = a \)
Answer: (a) \( \lim_{x \to a} f(x) \) is integer, (d) \( f(x) \) has local minima at \( x = a \)

Question. Let \( f(x) = \begin{cases} x^3 + x^2 - 10x & -1 \le x < 0 \\ \sin x & 0 \le x < \pi/2 \\ 1 + \cos x & \pi/2 \le x \le \pi \end{cases} \) then \( f(x) \) has
(a) local maximum at \( x = \pi/2 \)
(b) local minima at \( x = \pi/2 \)
(c) absolute minima at \( x = 0, \pi \)
(d) absolute maxima at \( x = \pi/2 \)
Answer: (a) local maximum at \( x = \pi/2 \), (c) absolute minima at \( x = 0, \pi \)

Question. The sum of the legs of a triangle is 9 cm. When the triangle rotates about one of the legs, a cone results which has the maximum volume. Then
(a) slant height of such a cone is \( 3\sqrt{5} \)
(b) maximum volume of the cone is \( 32\pi \)
(c) curved surface of the cone is \( 18\sqrt{5}\pi \)
(d) semi vertical angle of cone is \( \tan^{-1} \sqrt{2} \)
Answer: (a) slant height of such a cone is \( 3\sqrt{5} \), (c) curved surface of the cone is \( 18\sqrt{5}\pi \)

Question. The function \( f(x) = \sin x - x \cos x \) is
(a) maximum or minimum for all integral multiple of \( \pi \)
(b) maximum if \( x \) is an odd positive or even negative integral multiple of \( \pi \)
(c) minimum if \( x \) is an even positive or odd negative integral multiple of \( \pi \)
(d) None of the options
Answer: (a) maximum or minimum for all integral multiple of \( \pi \), (b) maximum if \( x \) is an odd positive or even negative integral multiple of \( \pi \), (c) minimum if \( x \) is an even positive or odd negative integral multiple of \( \pi \)

Question. The curve \( y = \frac{x + 1}{x^2 + 1} \) has
(a) \( x = 1 \), the point of inflection
(b) \( x = -2 + \sqrt{3} \), the point of inflection
(c) \( x = -1 \), the point of minimum
(d) \( x = -2 - \sqrt{3} \), the point of inflection
Answer: (a) \( x = 1 \), the point of inflection, (b) \( x = -2 + \sqrt{3} \), the point of inflection, (d) \( x = -2 - \sqrt{3} \), the point of inflection

Question. If the derivative of an odd cubic polynomial vanishes at two different values of 'x' then
(a) coefficient of \( x^3 \) & \( x \) in the polynomial must be same in sign
(b) coefficient of \( x^3 \) & \( x \) in the polynomial must be different in sign
(c) the values of 'x' where derivative vanishes are closer to origin as compared to the respective roots on either side of origin
(d) the values of 'x' where derivative vanishes are far from origin as compared to the respective roots on either side of origin
Answer: (b) coefficient of \( x^3 \) & \( x \) in the polynomial must be different in sign, (c) the values of 'x' where derivative vanishes are closer to origin as compared to the respective roots on either side of origin

Question. Let \( f(x) = \ln(2x - x^2) + \sin \frac{\pi x}{2} \). Then
(a) graph of f is symmetrical about the line \( x = 1 \)
(b) graph of f is symmetrical about the line \( x = 2 \)
(c) maximum value of f is 1
(d) minimum value of f does not exist
Answer: (a) graph of f is symmetrical about the line \( x = 1 \), (c) maximum value of f is 1, (d) minimum value of f does not exist

Question. The maximum and minimum values of \( y = \frac{Ax^2 + 2Bx + C}{ax^2 + 2bx + c} \) are those for which
(a) \( ax^2 + 2bx + c - y(Ax^2 + 2Bx + C) \) is equal to zero
(b) \( ax^2 + 2bx + c - y(Ax^2 + 2Bx + C) \) is perfect square
(c) \( \frac{dy}{dx} = 0 \) and \( \frac{d^2y}{dx^2} \neq 0 \)
(d) \( ax^2 + 2bx + c - y(Ax^2 + 2Bx + C) \) is not a perfect square
Answer: (b) \( ax^2 + 2bx + c - y(Ax^2 + 2Bx + C) \) is perfect square, (c) \( \frac{dy}{dx} = 0 \) and \( \frac{d^2y}{dx^2} \neq 0 \)

Question. Maximum and minimum values of the function, \( f(x) = \frac{2 - x}{\pi} \cos (\pi(x + 3)) + \frac{1}{\pi^2} \sin (\pi (x + 3)) \) \( 0 < x < 4 \) occur at
(a) \( x = 1 \)
(b) \( x = 2 \)
(c) \( x = 3 \)
(d) \( x = \pi \)
Answer: (a) \( x = 1 \), (c) \( x = 3 \)

Question. If \( f(x) = \log (x - 2) - \frac{1}{x} \), then
(a) \( f(x) \) is M.I. for \( x \in (2, \infty) \)
(b) \( f(x) \) is M.I. for \( x \in [-1, 2] \)
(c) \( f(x) \) is always concave downwards
(d) \( f^{-1} (x) \) is M.I. wherever defined
Answer: (a) \( f(x) \) is M.I. for \( x \in (2, \infty) \), (c) \( f(x) \) is always concave downwards, (d) \( f^{-1} (x) \) is M.I. wherever defined

Advanced Subjective Questions

Question. Show that for each \(a > 0\) the function \(e^{-ax} \cdot x^a\) has a maximum value say \(F(a)\), and that \(F(x)\) has a minimum value, \(e^{-e/2}\).
Answer: \(f(x) = e^{-ax} \cdot x^a\)
\(f'(x) = -a e^{-ax} \cdot x^a + a^2 e^{-ax} \cdot x^{a-1} = 0\)
\(f'(x) = 0 \Rightarrow x = a\)
min value at \(x = a\)
\(F(a) = e^{-a^2} \cdot a^{a^2}\)
\(F(x) = e^{-x^2} \cdot x^{x^2}\)
\(F'(x) = -2x e^{-x^2} \cdot x^{x^2} + e^{-x^2} x^{x^2} (2x \ln x + x) = 0\)
\(2x \ln x - x = 0\)
\(x = 0, \ln x = 1/2\)
\(x = \sqrt{e}\)
min. value \(= e^{-e} \left(\sqrt{e}\right)^e = e^{-e} \cdot e^{e/2} = e^{-e/2}\)

Question. For \(a > 0\), find the minimum value of the integral \(\int_{0}^{1/a} (a^3 + 4x - a^5 x^2)e^{ax} dx\).
Answer: \(f(a) = \int_{0}^{1/a} (a^3 + 4x - a^5 x^2)e^{ax} dx\)
differentiate with respect to 'a' using leibnitz and equate to zero to get critical points and hence find minimum value.

Question. Consider the function \(f(x) = \begin{cases} \sqrt{x} \ln x & \text{when } x > 0 \\ 0 & \text{for } x = 0 \end{cases}\)
(a) Find whether f is continuous at x = 0 or not.
(b) Find the minima and maxima if they exist.
(c) Does f'(0) ? Find \(\lim_{x \to 0} f'(x)\)
(d) Find the inflection points of the graph of \(y = f(x)\).
Answer: (a) \(f(0) = 0\)
\(\lim_{x \to 0^+} \sqrt{x} \ln x = \lim_{x \to 0^+} \frac{\ln x}{x^{-1/2}}\) (By L'Hospital)
\(= \lim_{x \to 0^+} \frac{1/x}{-\frac{1}{2} x^{-3/2}} = 0\)
f is continuous at \(x = 0\)
(b) \(f'(x) = \frac{1}{2\sqrt{x}} \ln x + \frac{1}{\sqrt{x}} = 0\)
\(\Rightarrow \ln x + 2 = 0\)
\(\Rightarrow x = e^{-2}\)
(c) \(f'(0^+) = \lim_{h \to 0} \frac{\sqrt{h} \ln h - 0}{h}\)
\(\lim_{h \to 0} \frac{\ln h}{\sqrt{h}} \to -\infty\)
DNE
\(\lim_{h \to 0} f'(x)\) is also DNE
(d) \(f'(x) = \frac{\ln x + 2}{2\sqrt{x}}\)
\(f''(x) = \frac{2\sqrt{x} \left(\frac{1}{x}\right) - (\ln x + 2) \frac{1}{\sqrt{x}}}{4x}\)
\(f''(x) = 0 \Rightarrow x = 0\)

Question. Consider the function \(y = f(x) = \ln (1 + \sin x )\) with \(-2\pi \le x \le 2\pi\). Find
(a) the zeroes of \(f(x)\)
(b) inflection points if any on the graph
(c) local maxima and minima of \(f(x)\)
(d) asymptotes of the graph
(e) sketch the graph of \(f(x)\) and compute the value of the definite integral \(\int_{-\pi/2}^{\pi/2} f(x) dx\).
Answer: \(f(x) = \ln (1 + \sin x)\)
(a) zeroes of \(f(x)\)
where \(\sin x = 0\) \(\forall x \in [-2\pi, 2\pi]\)
\(\Rightarrow x = -2\pi, -\pi, 0, \pi, 2\pi\)
(b) \(f'(x) = \frac{\cos x}{1 + \sin x}\)
\(f''(x) = \frac{(1 + \sin x)(-\sin x) - \cos^2 x}{(1 + \sin x)^2}\)
\(= \frac{-\sin x - 1}{(1 + \sin x)^2} \ne 0\)
So, no inflection point.
(c) \(f'(x) = 0\)
\(\Rightarrow x = \frac{\pi}{2}, -\frac{3\pi}{2}\)
(d) Asymptotes where \(\sin x = -1\)
\(\Rightarrow x = -\frac{\pi}{2}, \frac{3\pi}{2}\)
(e) \(I = \int_{-\pi/2}^{\pi/2} \ln (1 + \sin x)dx\)
king & add
\(2I = \int_{-\pi/2}^{\pi/2} \ln (1 - \sin^2 x)dx\)
\(I = \int_{0}^{\pi/2} \ln \cos^2 x dx = 2 \int_{0}^{\pi/2} \ln \cos x dx\)
\(= 2 \int_{0}^{\pi/2} \ln \sin x dx = 2 \int_{0}^{\pi/2} \ln \sin x dx\)
\(I = -\pi \ln 2\)

Question. Find the set of value of m for the cubic \(x^3 - \frac{3}{2}x^2 + \frac{5}{2} = \log_{1/4}(m)\) has 3 distinct solutions.
Answer: \(f(x) = x^3 - \frac{3}{2} x^2 + \frac{5}{2} + \log_4 m\)
\(f'(x) = 3x^2 - 3x = 0\)
\(x = 0, 1\)
\(f(0) \cdot f(1) < 0\)
\(\left(\frac{5}{2} + \log_4 m\right) \left(1 - \frac{3}{2} + \frac{5}{2} + \log_4 m\right) < 0\)
\(\left(\frac{5}{2} + \log_4 m\right) (2 + \log_4 m) < 0\)
\(-\frac{5}{2} < \log_4 m < -2\)
\(4^{-5/2} < m < 4^{-2}\)
\(\frac{1}{32} < m < \frac{1}{16}\)

Question. The value of 'a' for which \(f(x) = x^3 + 3(a - 7)x^2 + 3(a^2 - 9)x - 1\) have a positive point of maximum lies in the interval \((a_1, a_2) \cup (a_3, a_4)\). Find the value of \(a_2 + 11a_3 + 70a_4\).
Answer: \(f'(x) = 3x^2 + 6(a - 7)x + 3(a^2 - 9) = 0\)
must have both roots positive
(i) \(D > 0 \Rightarrow 36(a - 7)^2 - 36(a^2 - 9) > 0 \Rightarrow a < \frac{29}{7}\)
(ii) \(f(0) > 0 \Rightarrow a \in (-\infty, -3) \cup (3, \infty)\)
(iii) \(-\frac{b}{2a} > 0 \Rightarrow \frac{-6(a - 7)}{3} > 0 \Rightarrow a < 7\)
\(\therefore a \in (-\infty, -3) \cup \left(3, \frac{29}{7}\right)\)
\(\Rightarrow a_2 = -3, a_3 = 3, a_4 = \frac{29}{7}\)
\(a_2 + 11a_3 + 70a_4 = 320\)