jee-advanced 2014 Q43

jee-advanced · India · paper1 Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be given by $$f(x) = \int_{\frac{1}{x}}^{x} e^{-\left(t + \frac{1}{t}\right)} \frac{dt}{t}$$ Then
(A) $f(x)$ is monotonically increasing on $[1, \infty)$
(B) $f(x)$ is monotonically decreasing on $(0,1)$
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$, for all $x \in (0, \infty)$
(D) $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be given by
$$f(x) = \int_{\frac{1}{x}}^{x} e^{-\left(t + \frac{1}{t}\right)} \frac{dt}{t}$$
Then\\
(A) $f(x)$ is monotonically increasing on $[1, \infty)$\\
(B) $f(x)$ is monotonically decreasing on $(0,1)$\\
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$, for all $x \in (0, \infty)$\\
(D) $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$

Paper Questions

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