Question

please solve this. if don't know leave it ​

Answer 1

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

Hence we get required answer.