Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 31 maths

Explain solution RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 31 maths

Answers (1)

Answer:

f(x) \text { is strictly increasing on }(-\frac{\pi }{2},0) \text { and strictly decreasing on }(0,\frac{\pi }{2})

Given:

f(x)=log\: cos\: x

To prove:

\text { We have to prove that } f(x) \text { is strictly increasing on }(-\frac{\pi }{2},0) \text { and strictly decreasing on }(0,\frac{\pi }{2})

Hint:

  1. for f(x) to be increasing we must have f'(x)>0
  2. for f(x) to be decreasing we must have f'(x)<0

Solution:

Given

f(x)=log\: cos\: x

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}[\log \cos x] \\ &\Rightarrow f^{\prime}(x)=\frac{1}{\cos x} \times(-\sin x) \\ &\Rightarrow f^{\prime}(x)=-\tan x \\ &\text { In interval }\left(0, \frac{\pi}{2}\right), \tan x>0 \\ &\Rightarrow-\tan x<0 \end{aligned}

By applying negative sign change comparison sign

\therefore f^{\prime}(x)<0 \text { on }\left(0, \frac{\pi}{2}\right)

\text { Hence } f(x) \text { is strictly decreasing on }(0,\frac{\pi }{2})

\begin{aligned} &\text { In interval }\left(\frac{\pi}{2}, \pi\right), \tan x<0 \\ &\Rightarrow-\tan x>0 \end{aligned}

By applying negative sign change comparison sign

\therefore f^{\prime}(x)>0 \text { on }\left(\frac{\pi}{2}, \pi\right)

\text { Hence } f(x) \text { is strictly increasing on }(\frac{\pi }{2},\pi )

Posted by

Gurleen Kaur

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 12 board exams 2025 to have 50% competency-based questions; no change in 10th
anam-khan

CBSE Class 12 board exams 2024 over; expected result date, trends of last 10 years

Latest Question