Question

find the intervals in which the function f(x)=tan x -4x where x belongs to (0, pie/2) is strictly increasing b) strictly decreasing​

Answer 1

Given : f(x)= tan x -4x where x belongs to (0, π/2)

To Find : intervals in which the function is strictly increasing b) strictly decreasing​

Solution:

f(x)=tan x -4x

f'(x) = sec²x  - 4

for strictly  increasing interval f'(x)  > 0 =>  sec²x  - 4 > 0

=> sec²x > 4

=> Secx > 2    Secx is  + ve in  ( 0 , π/2)

Hence π/3 <  x  <  π/2      

for strictly decreasing interval f'(x)  < 0 =>  sec²x  - 4 < 0

=> sec²x <  4

=> 0 < Secx  < 2  as secx  is + ve in  ( 0 , π/2)

Hence strictly decreasing for  0 <  x < π/3

strictly decreasing for  0 < x < π/3

strictly increasing for   π/3 <  x  <  π/2

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