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) = x2 − x + 1.

f'(x) = 2x-1

Now, f'(x) = 0$\implies$ x = $\frac 12$.

The point $\frac 12$divides the interval (−1, 1) into two disjoint intervals

i.e., (-1,$\frac 12$) and ($\frac 12$,1).

Now, in interval (-1,$\frac 12$), f'(x) = 2x-1<0.

Therefore, f is strictly decreasing in interval (-1,$\frac 12$).

However, in interval ($\frac 12$,1), f'(x) = 2x-1>0.

Therefore, f is strictly increasing in interval ($\frac 12$,1).

Hence, f is neither strictly increasing nor decreasing in interval (−1, 1)