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(๐ฅ) = tan ๐ฅ โ ๐ฅ
Finding ๐^โฒ (๐)
๐^โฒ (๐ฅ)=ใ๐๐๐ใ^๐ ๐ โ๐
Now, we need to check if ๐(๐) is increasing or decreasing
Checking sign of ๐^โฒ (๐)
๐^โฒ (๐ฅ)=ใ๐๐๐ใ^๐ ๐ โ๐
Now,
๐ฌ๐๐ ๐ ๐ (โโ,โ๐]โช[๐,โ)
So,
ใ๐๐๐ใ^๐ ๐ ๐ [1,โ)
Therefore, we can write this as
๐โคใ๐๐๐ใ^๐ ๐โคโ
1โ๐โคใ๐ ๐๐ใ^2 ๐ฅโ๐โคโโ๐
0โคใ๐๐๐ใ^๐ ๐โ๐โคโ
Thus,
ใ๐ ๐๐ใ^2 ๐ฅโ1โฅ0
โด ๐^โฒ (๐)>๐ for all values of x
Hence, ๐(๐) always increases.
So, the correct answer is (A)

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.