Review question

# How many regions are created by the graphs $y=x^3, y=x^4$ and $y=x^5$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5685

## Solution

Into how many regions is the plane divided when the following equations are graphed, not considering the axes?

$y=x^3$

$y=x^4$

$y=x^5$

1. $6$, (b) $7$, (c) $8$, (d) $9$, (e) $10$.

A careful sketch seems like our best plan, noting that all three curves go through $(0,0)$ and $(1,1)$, and that $x^3>x^4>x^5$ on the interval $0<x<1$.

We also have that $y=x^3$ and $y=x^5$ both go through $(-1,-1)$, and $x^3 < x^5 < 0$ on the interval $-1<x<0$.

So here is our sketch, without including the axes;

We have nine regions, and so the answer is (d).