Critical Points

A critical point is simply a spot where the graph is either perfectly flat for a moment or has a sharp corner. They matter because every peak and every valley of a function has to happen at one, so they're the short list of places to look when you want the biggest or smallest value. The definition "$f'(c) = 0$ or $f'(c)$ does not exist" is just those two cases: $f'(c) = 0$ (the tangent is flat) and "does not exist" (there's no tangent, like at a corner or cusp). The extra condition that $f(c)$ is defined just means the point has to actually be on the graph. That's all there is to it.

Think about walking along a smooth graph. Just before the top of a hill you're going up, so the slope is positive. Just after, you're going down, so it's negative. Right at the top, the slope has to pass through zero. That's why peaks and valleys on smooth curves have flat tangents. The other way to get a peak is a sharp point, like the bottom of $|x|$, where there's no slope at all. Either way, it's a critical point.

Finding them:

1. Differentiate and simplify into a single fraction, factored if possible.
2. Solve $f'(x) = 0$, which means the numerator equals zero.
3. Find where $f'$ is undefined (the denominator is zero), and keep those points only if $f$ itself is defined there.

A couple of things help. $e^{\text{anything}}$ is never zero, so it can be divided out. And rational exponents like $x^{2/3}$ often give derivatives that are undefined at $x = 0$, because the exponent turns negative after differentiating.

A common mistake. It's tempting to think every critical point is a peak or a valley, since that's what we were looking for. It isn't: $x^3$ has a flat tangent at 0, but it keeps climbing right through it. Critical points are candidates, not answers. The next few lessons show how to sort them out.

Engineers find the stress peaks in a beam, and the angle of a solar panel that collects the most light, by locating critical points.