The given function f is defined as,
f( x )=logsinx
The derivative of the function f is given as,
f ′ ( x )= df( x ) dx = d( logsinx ) dx = 1 sinx ×cosx =cotx
For the given interval ( 0, π 2 ),
cotx>0
Hence, f ′ ( x )>0.
As f ′ ( x )>0 in the interval ( 0, π 2 ), therefore f is strictly increasing in the interval ( 0, π 2 ).
For the given interval ( π 2 ,π ),
cotx<0
Hence, f ′ ( x )<0.
As f ′ ( x )<0 in the interval ( π 2 ,π ), therefore f is strictly decreasing in the interval ( π 2 ,π ).
Hence, it is proved that the function f is strictly increasing in interval ( 0, π 2 ) and strictly decreasing in the interval ( π 2 ,π ).