Question

Show that the function $x^2-x+1$ is neither increasing nor decreasing on $(0,1)$.

Answer 1

Given,

the function $f\left(x\right)=x^2-x+1$

$f\left(x\right)=x^2-x+1$

$f^{\prime }\left(x\right)=2x-1$

$f^{\prime }\left(x\right)>0$, $\forall x\in (\frac{1}{2},1)$ $[\because f^{\prime }(x)>0\Rightarrow$ strictly increasing$]$

$f^{\prime }\left(x\right)<0,\forall x\in (0,\frac{1}{2})$ $[\because f^{\prime }(x)<0\Rightarrow$ strictly decreasing$]$

clearly,

we can see that

$f\left(x\right)$ is strictly increasing in the interval $(\frac{1}{2},1)$

$f\left(x\right)$ is strictly decreasing in the interval $(0,\frac{1}{2})$

$\therefore f\left(x\right)$ is neither increasing nor decreasing on the whole interval $(0,1)$.