, ∈ as ≤ ⇒ ≤ and x < ′ for all π π ∈ as ⇒ Therefore, f is increasing in , and decreasing in π π . Fig . Also, the given function is continuous at x = and . Therefore, by Theorem , f is increasing on , π and decreasing on π π .
Example Find the intervals in which the function f given by f ( x ) = sin x + cos x , ≤ x ≤ π is increasing or decreasing. Solution We have f ( x ) = sin x + cos x , f ′ ( x ) = cos x – sin x Now x = ′ gives sin x = cos x which gives that , π as ≤ ≤π The points and divide the interval [ , π ] into three disjoint intervals, namely, , , π and , Note that if , , ′ > ∈ ∪ f is increasing in the intervals , and , Also ′ ∈ if f is decreasing in Fig . Interval Sign of ′ Nature of function , > f is increasing < f is decreasing , > f is increasing EXERCISE . .
Show that the function given by f ( x ) = x + is increasing on R . . Show that the function given by f ( x ) = e x is increasing on R . .
Show that the function given by f ( x ) = sin x is