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)