Question

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

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

Answer 1

The correct option is C

Neither increasing nor decreasing

Determine if the function is increasing or decreasing.

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

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

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

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

Hence, option C is the correct answer.