Question

In the interval $[0,1]$, the function $x^2-x+1$ is

• A. Increasing
• B. Decreasing
• C. Neither increasing nor decreasing
• D. None

Answer 1

The correct option is C

Neither increasing nor decreasing

Determine the increasing or decreasing property of the function $x^2-x+1$

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

$$\begin{gathered} \Rightarrow f\text{'}\left(x\right)=2x-1\text{[Differentiting w.r.t.}x] \\ \Rightarrow f\text{'}(x)>0,\forall x\in \left(\frac{1}{2},1\right)\left[\because f\text{'}\left(x\right)>0\Rightarrow \text{strictly increasing}\right] \end{gathered}$$

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

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

Hence, option C is the correct answer.