Question

Prove that the function $f$ defined by $f\left(x\right)=x^2-x+1$ is neither increasing nor decreasing in $(-1,1)$. Hence find the intervals in which $f\left(x\right)$ is strictly increasing and strictly decreasing.

Answer 1

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

$\frac{dy}{dx}=2x-12x-1>0$

$x>\frac{1}{2}increasingx\in (\frac{1}{2},\infty )$

$2x-1<0x<\frac{1}{2}drcreasingx\in (-\infty \frac{1}{2})$

hence $f\left(x\right)$ is nither decreasing nor increasing in$(-1,1)$