Question

011. If the function ()
K sin x+2cosx
is
sin -+ cos2
increasing for all values of x then
A K<1
B K> 1
C K<2
D K>2​

Answer 1

Answer:

Step-by-step explanation:

ANSWER

The given function is:

f(x)=

sinx+cosx

Ksinx+2cosx

​

Differentiating once w.r.t x and and making it greater than 0 to make it monotonically increasing we get,

⇒f

′

(x)=

(sinx+cosx)

2

(Kcosx−2sinx)(sinx+cosx)−(cosx−sinx)(Ksinx+2cosx)

​

⇒f

′

(x)=

(sinx+cosx)

2

Ksinxcosx+Kcos

2

x−2sin

2

x−2sinxcosx−Ksinxcosx−2cos

2

x+Ksin

2

x+2sinxcosx

​

⇒f

′

(x)=

(sinx+cosx)

2

Kcos

2

x+Ksin

2

x−2cos

2

x−2sin

2

x

​

⇒f

′

(x)=

(sinx+cosx)

2

K−2

​

Now, f

′

(x)>0

⇒

(sinx+cosx)

2

K−2

​

>0

⇒K−2>0

⇒K>2 .....Answer