Math, asked by chauhanaayush2003, 7 months ago

6. Find the intervals in which the functionſgiven by
4 sin x - 2x - Xcosx/

2 + COS X
is (i) strictly increasing (ii) strictly decreasing.​

Answers

Answered by samirpanchal0092
6

Answer:

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MATHS

Find the intervals in which the function f(x)=

2+cosx

4sinx−2x−xcosx

is (i) increasing

(ii) decreasing

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ANSWER

f(x)=

2+cosx

4sinx−2x−xcosx

=

2+cosx

4sinx

2+cosx

2x+xcosx

=

2+cosx

4sinx

(2+cosx)

x(2+cosx)

=

2+cosx

4sinx

−x

Now let's find out f

(x)

f

(x)=

(2+cosx)

2

4cosx(2+cosx)+sinx(4sinx)

−1

=

(2+cosx)

2

8cosx+4(cos

2

x+sin

2

x)

−1

=

(2+cosx)

2

8cosx+4−(2+cosx)

2

=

(2+cosx)

2

8cosx+4−4−4cosx−cos

2

x

=

(2+cosx)

2

cosx(4−cosx)

Now,(2+cosx)

2

>0∀xϵR

Similarly,4−cosx>0∀xϵR

as cosxϵ[−1,1]

⇒4−cosxϵ[3,5]>0

∴ the function f(x) will be increasing if f

(x)>0⇒cosx>0 and decreasing if f

(x)<0⇒cosx<0

(i) Increasing

f

(x)>0

⇒cosx>0

∴xϵ(0,

4

π

)∪(

4

,2π)

(ii) Decreasing

f

(x)<0

⇒cosx<0

∴xϵ(

4

π

,π)∪(π,

2

)

Answered by xxzara42xx
3

Answer:

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