Question

Prove that the function f given by f ( x ) = x 2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

Answer 1

The given function is f( x )= x 2 −x+1.

Differentiate the function with respect to x.

f ′ ( x )= d dx ( x 2 −x+1 ) =2x−1

Substitute f ′ ( x ) = 0,

2x−1=0 2x=1 x= 1 2

Now, the point x= 1 2 divides the domain ( −1,1 ) into two disjoint intervals given by,

( −1, 1 2 ) ( 1 2 ,1 )

In the interval ( −1, 1 2 ).

f ′ ( x )<0 2x−1<0

The function is strictly decreasing in the domain ( −1, 1 2 ).

In the interval ( 1 2 ,1 ).

f ′ ( x )>0 2x−1>0

The function is strictly increasing in the domain ( 1 2 ,1 ).

Thus, the function f( x )= x 2 −x+1 is neither strictly increasing nor strictly decreasing in the interval ( −1,1 ).