Let the function g:(-∞, ∞) arrow(-(π)/(2), (π)/(2)) be given by g(u)=2 ^ -1 (e^u)-(π)/(2) . Then, g is

Correct answer: C. odd and is strictly increasing in (-∞, ∞) g(u)=2 ^ -1 (e^u)-(π)/(2) for u ∈(-∞, ∞) and aligned g(-u) & =2 ^ -1 (e^ -u )-(π)/(2) & =2 ( ^ -1 (e^u))-(π)/(2) & =2 ((π)/(2)- ^ -1 (e^u))-(π)/(2) & =(π)/(2)-2 ^ -1 (e^u)=-g(u) aligned g(-u)=-g(u) g(u) is an odd function.

JEE Advanced 2008Paper 2

Exam: JEE Advanced
Year: 2008
Paper: Paper 2
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2008 · Paper 2 · Mathematics · Application of Derivatives

Let the function g:(-∞, ∞) arrow(-(π)/(2), (π)/(2)) be given by...

Let the function

g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

be given by

g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}

. Then,

g

Options

Aeven and is strictly increasing in $(0, \infty)$
Bodd and is strictly decreasing in $(-\infty, \infty)$
Codd and is strictly increasing in $(-\infty, \infty)$
DNeither even nor odd, but is strictly increasing in $(-\infty, \infty)$

Correct answer

C. odd and is strictly increasing in $(-\infty, \infty)$

Step-by-step solution

\therefore \quad g(u)=2 \tan ^{-1}

\left(e^u\right)-\frac{\pi}{2}

for

\quad u \in(-\infty, \infty)

and

\begin{aligned}g(-u) & =2 \tan ^{-1}\left(e^{-u}\right)-\frac{\pi}{2}\\ & =2 \left(\cot ^{-1}\left(e^u\right)\right)-\frac{\pi}{2}\\ & =2 \left(\frac{\pi}{2}-\tan ^{-1}\left(e^u\right)\right)-\frac{\pi}{2}\\ & =\frac{\pi}{2}-2 \tan ^{-1}\left(e^u\right)=-g(u) \end{aligned}

\therefore \quad g(-u)=-g(u)\Rightarrow g(u)$ is an odd function.

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the function f: (0, ∞) arrow (-∞, ∞) given by f(x) = x \, _e(x) - x + 1 . Then… JEE Advanced 2026 Let P be the point on the parabola y = x^2 such that the slope of the tangent to the… JEE Advanced 2026 Let f: ℝ arrow ℝ be a differentiable function such that f((x+y)/(3)) = (f(x) + f(y))/(3)… JEE Main 2026 _ 0 ≤ x ≤ π (16 ((x)/(2)) ^3((x)/(2))) is equal to: JEE Main 2026 Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x → 0… JEE Main 2026 The number of critical points of the function f(x) = cases |( x)/(x)|, & x ≠ 0 1, & x = 0… JEE Main 2026 Let f be a differentiable function satisfying f(x)=1-2 x+∫_ 0 ^ x e^ (x-t) f(t) dt, x ∈ R… JEE Main 2026 Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by… JEE Main 2026 Let (2 α, α) be the largest interval in which the function f(t)= |t+1| t^ 2 , t 2 , is… JEE Main 2026 Let f(x)=x^ 2025 -x^ 2000 , x ∈[0,1] and the minimum value of the function f(x) in the… JEE Main 2026 Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and… JEE Main 2026 Let f: R arrow R be a twice differentiable function such that the quadratic equation f(x)… JEE Main 2026 Let ℝ denote the set of all real numbers. Define the function f: R arrow R by f(x)=\… JEE Advanced 2025 Let ℝ denote the set of all real numbers. For a real number x , let [x] denote the… JEE Advanced 2025 Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be defined by f(x)= cases (6… JEE Advanced 2025 Let the function f(x)=(x)/(3)+(3)/(x)+3, x ≠ 0 be strictly increasing in (-∞, α_1) U(α_2… JEE Main 2025 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs