Increasing And Decreasing Functions

Differentiating on both sides

⇒ g‘(x)=cos⁡x−sin⁡x>0 in
⇒ g(x) is increasing in
⇒ Option (A) is wrong.

Now,
g”(x)=−sin⁡x−cos⁡x<0 in
⇒ g(x) is increasing in
⇒ Option (B) is wrong.

Let h(x) = g(x) + g′(x)
⇒ ℎ‘(x)=g‘(x)+g”(x)=−2sin⁡x<0 in x∈
⇒ g + g' is decreasing in
⇒ Option (C) is wrong.

Let J(x) = g(x) – g′(x)
J‘(x)=g‘(x)−g”(x)=2cos⁡x>0 in
⇒ g – g′ is increasing in
⇒ Option (D) is correct.

So, the correct option is (D): g-g' is increasing in

The correct option is (D):