Application of Derivatives 2 Question 25
####25. Match the conditions/expressions in Column I with statements in Column II.
Let the functions defined in Column I have domain $(-\pi / 2, \pi / 2)$.
| Column I | |
|---|---|
| A. $x+\sin x$ | p. increasing |
| B. $\sec x$ | q. decreasing |
| r. neither increasing nor decreasing |
Analytical & Descriptive Questions
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Answer:
Correct Answer: 25. (c)
Solution:
- $\frac{d}{d x}(x+\sin x)=1+\cos x=2 \cos ^{2} \frac{x}{2}>0$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.
Therefore, $x+\sin x$ is increasing in the given interval. Therefore, $(\mathrm{A}) \rightarrow(\mathrm{p})$ is the answer.
Again, $\frac{d}{d x}(\sec x)=\sec x \tan x$ which is $>0$ for $0<x<\pi / 2$ and
$ <0 \text { for } \frac{-\pi}{2}<x<0 $
Therefore, sec $x$ is neither increasing nor decreasing in the given interval. Therefore, $(\mathrm{B}) \rightarrow(\mathrm{r})$ is the answer.
