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  • Explain solution RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 24 maths

Explain solution RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 24 maths

Answers (1)

Answer:

 f(x) is decreasing function on R

Given:

f(x)=tan^{-1}x-x

To prove:

We have to show that f(x) is decreasing function on R.

Hint:

A function f(x) to be decreasing if f ’(x) > 0

Solution:

We have

f(x)=tan^{-1}x-x

On differentiating both sides w.r.t x we get

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\tan ^{-1} x-x\right) \\ &=\frac{1}{1+x^{2}}-1,\left[\therefore \frac{d}{d x} \tan ^{-1} x=\frac{1}{1+x^{2}}\right] \\ &\Rightarrow f^{\prime}(x)=\frac{-x^{2}}{1+x^{2}} \end{aligned}

\begin{aligned} &\text { Now, } x \in R \\ &\Rightarrow x^{2}>0 \text { and } 1+x^{2}>0 \\ &\Rightarrow \frac{x^{2}}{1+x^{2}}>0 \\ &\Rightarrow \frac{-x^{2}}{1+x^{2}}<0 \end{aligned}

By applying negative sign change comparison sign

\begin{aligned} &\Rightarrow f'(x)< 0 \end{aligned}

Hence f(x) is decreasing function for x \inR

Posted by

Gurleen Kaur

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