Ty Moore

2022-11-22

Interval Notation for Increasing and Decreasing Intervals of a Function
This was brought up by another student in one of my pre-calculus classes.
The graph was a simple quadratic ${x}^{2}$. The teacher stated that the graph was decreasing from $\left(-\mathrm{\infty },0\right)$, and increasing from $\left(0,\mathrm{\infty }\right)$.
Why would zero not be included? i.e: decr. $\left(-\mathrm{\infty },0\right]$ and incr. $\left[0,\mathrm{\infty }\right)$

reinmelk3iu

Expert

Explanation:
Because for f(x) to be decreasing ${f}^{\prime }\left(x\right)<0$. And for increasing ${f}^{\prime }\left(x\right)>0$. But at $x=0$, ${f}^{\prime }\left(x\right)=0$ hence it's neither decreasing nor increasing at $x=0$

figoveck38

Expert

Explanation:
Generally the 0 is not included because the function is not decreasing (or increasing) at 0.
It would be accurate, however to say that $y={x}^{2}$ is non-increasing on the interval $\left(-\mathrm{\infty },0\right]$

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