Question

On which of the following intervals is the function f given by strictly decreasing? (A) (B) (C) (D) None of these

Answer 1

The given function f is defined as,

f( x )= x 100 +sinx−1

The derivative of f is given as,

f ′ ( x )= df( x ) dx = d( x 100 +sinx−1 ) dx =100 x 99 +cosx

In the given interval ( 0,1 ),

cosx>0 100 x 99 >0

Hence, f ′ ( x )>0.

As f ′ ( x )>0, therefore the given function f is strictly increasing in the interval ( 0,1 ).

In the given interval ( π 2 ,π ),

cosx<0 100 x 99 >0

As the value of function 100 x 99 is always greater than cosx, hence

f ′ ( x )>0

As f ′ ( x )>0, therefore the given function f is strictly increasing in the interval ( π 2 ,π ).

In the given interval ( 0, π 2 ),

cosx>0 100 x 99 >0

Hence, f ′ ( x )>0.

As f ′ ( x )>0, therefore the given function f is strictly increasing in the interval ( 0, π 2 ).

As the given function f is strictly increasing in all the given intervals therefore none of these is the correct option. Hence, the correct answer is D.