On which of the following intervals is the function x 100 + sin x - 1 decreasing?

Step-by-step solution

f x = x 100 + sin x − 1 ⇒ f ′ x = 100 x 99 + cos x If 0 < x < π 2 , then f ′ x > 0 , , therefore f x is increasing on 0 , π 2 . If 0 < x < 1 , then 100 x 99 > 0 and cos x > 0 [ ∵ x lies between 0 and 1 ] ⇒ f ′ x = 100 x 99 + cos x > 0 ⇒ f x is increasing on 0 , 1 . If π 2 < x < π , then 100 x 99 > 100 ∵ x > 1 ⇒ x 99 > 1 ⇒ 100 x 99 + cos x > 0 ∵ cos x ≥ − 1 ⇒ 100 x 99 + cos x > 99 ⇒ f ′ x > 0 ⇒ f x is increasing in π 2 , π .