QJEEAdv08Ma3200: JEE Advanced 2008 Mathematics - Class 12 JEEAdv08Ma3200 Solution: Let Which of the following is true?

Question

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Let $g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$

Which of the following is true?

A $g'(x)$ is positive on $\left( { - \infty ,0} \right)$ and negative on $\left( {0,\infty } \right)$
B $g'(x)$ is negative on $\left( { - \infty ,0} \right)$ and positive on $\left( {0,\infty } \right)$
C $g'(x)$ changes sign on both $\left( { - \infty ,0} \right)$ and $\left( {0,\infty } \right)$
D $g'(x)$ does not change sign on $\left( { - \infty ,0} \right)$

Accepted Answer

Given $g'(x) = \int\limits_0^{{e^x}} {{{f'(t)dt} \over {1 + {t^2}}}}$

$g''(x) = {{dg(x)} \over {d({e^x})}}.\,{{d({e^x})} \over {dx}}$

$= {{f'({e^x})} \over {1 + {e^{2x}}}}\,.\,{e^x}$

$= {{{e^x}} \over {1 + {e^{2x}}}}\,.\,{{2a({e^{2x}} - 1)} \over {{{({e^{2x}} + a{e^x} + 1)}^2}}} > 0$

when $x \in (0,\infty )$

$g'(x)$ < 0 when $x \in ( - \infty ,0)$

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