The function f (x) = tan x – x

(A)always increases  

(B) always decreases

(C) never increases

(D) sometimes increases and sometimes decreases.

 

This question is similar to Question 32 CBSE Class 12 Sample Paper 2021 Boards - Application of Derivatives

Slide86.JPG

Slide87.JPG
Slide88.JPG

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 22 The function f (x) = tan x โ€“ x always increases (B) always decreases (C) never increases (D) sometimes increases and sometimes decreases. Given ๐‘“(๐‘ฅ) = tan ๐‘ฅ โˆ’ ๐‘ฅ Finding ๐’‡^โ€ฒ (๐’™) ๐‘“^โ€ฒ (๐‘ฅ)=ใ€–๐’”๐’†๐’„ใ€—^๐Ÿ ๐’™ โˆ’๐Ÿ Now, we need to check if ๐’‡(๐’™) is increasing or decreasing Checking sign of ๐’‡^โ€ฒ (๐’™) ๐‘“^โ€ฒ (๐‘ฅ)=ใ€–๐’”๐’†๐’„ใ€—^๐Ÿ ๐’™ โˆ’๐Ÿ Now, ๐ฌ๐ž๐œ ๐’™ ๐ (โˆ’โˆž,โˆ’๐Ÿ]โˆช[๐Ÿ,โˆž) So, ใ€–๐’”๐’†๐’„ใ€—^๐Ÿ ๐’™ ๐œ– [1,โˆž) Therefore, we can write this as ๐Ÿโ‰คใ€–๐’”๐’†๐’„ใ€—^๐Ÿ ๐’™โ‰คโˆž 1โˆ’๐Ÿโ‰คใ€–๐‘ ๐‘’๐‘ใ€—^2 ๐‘ฅโˆ’๐Ÿโ‰คโˆžโˆ’๐Ÿ 0โ‰คใ€–๐’”๐’†๐’„ใ€—^๐Ÿ ๐’™โˆ’๐Ÿโ‰คโˆž Thus, ใ€–๐‘ ๐‘’๐‘ใ€—^2 ๐‘ฅโˆ’1โ‰ฅ0 โˆด ๐’‡^โ€ฒ (๐’™)>๐ŸŽ for all values of x Hence, ๐’‡(๐’™) always increases. So, the correct answer is (A)

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.