y = x³ − 3x
Here's y = x³ − 3x. It's an odd function, so it's symmetric about the origin, and it has two turning points where the slope is zero. Set the derivative 3x² − 3 equal to zero and you get x = −1 and x = 1.
- Local maximum at (−1, 2): the curve rises to here, then turns back down.
- Local minimum at (1, −2): the curve falls to here, then turns back up.
Between those two turns the curve crosses the x-axis at x = −√3, 0 and √3, roughly −1.73, 0 and 1.73. Outside the turning points it keeps climbing (for large positive x) or falling (for large negative x), like any cubic with a positive leading coefficient.
If you want to check the turns yourself, plug x = −1 and x = 1 into the original equation, then compare the values just before and after each point. The graph above marks both turning points.
