please solve this. if don't know leave it
Answers
Answer:
cos x
Step-by-step explanation:
Given :
f ( x ) = x / 1 + x tan x
We have find maximum value of x :
Diff. w.r.t. x :
f' ( x ) = [ ( 1 + x tan x ) ( x )' - ( x ) ( 1 + x tan x )' ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 + x tan x ) ( 1 ) - ( x ) ( 1 + x tan x )' ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 + x tan x ) - ( x ) ( 0 + ( x tan x )' ) ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 + x tan x ) - x ( x ( tan x )' + tan x ( x )' ) ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 + x tan x ) - ( x ) ( ( x sec² x ) + tan x ) ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 + x tan x - x² sec² x - x tan x ) ] / [ ( 1 + x tan x )² ]
f' ( x ) = [ ( 1 - x² sec² x ) ] / [ ( 1 + x tan x )² ]
Now for critical point :
f' ( x ) = 0
= > [ ( 1 - x² sec² x ) ] / [ ( 1 + x tan x )² ] = 0
= > ( 1 - x² sec² x ) = 0
= > x² sec² x = 1
= > x sec x = ± 1
For maximum :
= > x sec x =1
= > x = 1 / sec x
= > x = cos x