Question

Show that the function given by f ( x ) = sin x is (a) strictly increasing in (b) strictly decreasing in (c) neither increasing nor decreasing in (0, π)

Answer 1

The given function is f( x )=sinx.

Differentiate the function with respect to x.

f ′ ( x )=cosx

(a)

In the given range of ( 0, π 2 ),

cosx>0 f ′ ( x )>0 (1)

Thus, f( x ) is strictly increasing on ( 0, π 2 ).

(b)

In the given range of x∈( π 2 ,π ),

cosx<0 f ′ ( x )<0 (2)

Thus, f( x ) is strictly decreasing on ( π 2 ,π ).

(c)

In the given range of x∈( 0,π ), it is observed from equation (1) and equation (2) that, f( x ) is neither increasing nor decreasing.

Thus, f( x ) is neither increasing nor decreasing on ( 0,π ).