Application Of Derivatives Ques 93
93. Let $g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t$. Which of the following is true?
(a) $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
(b) $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
(c) $g^{\prime}(x)$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
(d) $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$
Show Answer
Answer:
Correct Answer: 93.(a)
Solution:
Formula:
Increasing and decreasing of a function:
- $\quad g^{\prime}(x)=\frac{f^{\prime}\left(e^{x}\right)}{1+\left(e^{x}\right)^{2}} \cdot e^{x}$
$ =2 a [\frac{e^{2 x}-1}{\left(e^{2 x}+a e^{x}+1\right)^{2}}] \quad (\frac{e^{x}}{1+e^{2 x}}) $
$ g^{\prime}(x)=0 \text {, if } e^{2 x}-1=0 \text {, i.e. } x=0 $
If
$ x<0, e^{2 x}<1 \Rightarrow g^{\prime}(x)<0 $
BETA
