Question

Prove that the function $f$ given by $f\left(x\right)=x^2-x+1$ is neither strictly increasing nor strictly decreasing on $(-1,1)$.

Answer 1

The given function is $f\left(x\right)=x^2-x+1$.
$\therefore f^{\prime }\left(x\right)=2x-1$
Now, $f^{\prime }\left(x\right)=0\Rightarrow x=\frac{1}{2}$.
The point $\frac{1}{2}$ divides the interval $(1,1)$ into two disjoint intervals i.e., $(-1,\frac{1}{2})$ and $(\frac{1}{2},1)$
Now, in interval $(-1,\frac{1}{2}),f^{\prime }\left(x\right)=2x-1<0$.
Therefore, $f$ is strictly decreasing in interval $(-1,\frac{1}{2})$.
However, in interval $(\frac{1}{2},1),f^{\prime }\left(x\right)>0\Rightarrow$. $f$ is increasing.
Hence, $f$ is neither strictly increasing nor decreasing in interval $(-1,1).$