Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Application of Derivatives
–1 vote

Show that $f(x)=2x+\cot^{-1}x+log\bigg(\sqrt {(1+x^2)}-x\bigg)$ is increasing in R.

Can you answer this question?

1 Answer

0 votes
  • Let $f(x)$ be a function defined on $(a,b)$. If $f'(x) >0$ for all $x \in (a,b)$ except for a finite number of points, where $ f'(x)>0$, then $f(x)$ is increasing on $(a,b)$
Step 1
$f(x)=2x+ \cot^{-1}x+ \log ( \sqrt{1+x^2}-x)$
differentiating w.r.t $X$ we get,
$f'(x)=2-\large\frac{1}{1+x^2}+\large\frac{1}{\sqrt{1+x^2-x}} \bigg( \large\frac{1}{2\sqrt{1+x^2}}.2x-1 \bigg)$
$ = 2-\large\frac{1}{1+x^2}+ \bigg(\large\frac{2x}{2\sqrt{1+x^2}}-1 \bigg) \large\frac{1}{\sqrt{1+x^2}-x}$
On simplifying we get
$ \large\frac{2x^2+x+2}{2x^2-x+2}- \large\frac{1}{\sqrt{1+x^2}-x} >0$
It is clear that for all values of $x$ the above equation gives only a positive value. Hence the function is increasing in $R$
answered Aug 4, 2013 by thanvigandhi_1

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App