Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise 16.1 , question 2 maths textbook solution

please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise 16.1 , question 2 maths textbook solution

Answers (1)

Answer:

             f(x)  is decreasing on(0,\infty )if 0<a<1

Hint:

  1. f(x)is increasing on(a,b), if the values of f(x) increase with the increase in the values of x.
  2. f(x)is decreasing on(a,b), if the values of f(x) decrease with the increase in the values of x.

Given:

Here given that,

               f(x)=\log _{a} x

To prove:

               Function f(x)=\log _{a} x is increasing on (0,\infty ), if a>1 and decreasing on (0,\infty ) , if 0<a<1

Solution:

Here we have two cases:

Case I: when a>1

Let us consider x_{1},x_{2}\epsilon (0,\infty )

Such that x_{1}<x_{2}
\begin{array}{ll} \Rightarrow & \log _{a} x_{1}<\log _{a} x_{2} \\ \Rightarrow & f\left(x_{1}\right)<f\left(x_{2}\right) \end{array}

Thus, f(x) is increasing in(0,\infty )if a>1

Case II: when 0<a<1

We have f(x)=\log _{a} x

We know that \log _{a} x=\frac{\log x}{\log a}

Therefore, f(x)=\frac{\log x}{\log a}

When a<1, then a<0

Let x_{1}<x_{2}

Taking log on both sides,

\begin{aligned} &\Rightarrow \quad & \log x_{1}<\log x_{2} \\ &\Rightarrow & \frac{\log x_{1}}{\log a}>\frac{\log x_{2}}{\log a} \\ &\Rightarrow & f\left(x_{1}\right)>f\left(x_{2}\right) \end{aligned}

 

Thus, f(x) is decreasing on \left ( 0,\infty \right ) if 0<a<1

 

Posted by

Info Expert 29

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
anam-khan

CBSE Class 12 accountancy board exam analysis 2024 out; paper rated easy, balanced

Latest Question