The First Derivative Test

"$f'$ changes from positive to negative at $c$" means that $f' > 0$ on some interval $(c - \delta, c)$ and $f' < 0$ on some interval $(c, c + \delta)$.

The proof uses the Mean Value Theorem. Suppose $f' > 0$ on $(c - \delta, c)$ and $f' < 0$ on $(c, c + \delta)$. Take any $x$ in $(c - \delta, c)$. Then $f$ is continuous on $[x, c]$ and differentiable on $(x, c)$, so for some $\xi$ in $(x, c)$,

So $f(x) < f(c)$. The same argument on $[c, x]$, for $x$ in $(c, c + \delta)$, gives $f(x) - f(c) = f'(\xi)(x - c) < 0$. Thus $f(c)$ is larger than every other value of $f$ on $(c - \delta, c + \delta)$, a relative maximum. The minimum case is the same with the signs reversed.

The proof needs $f$ continuous on $[x, c]$, so the test asks for continuity at $c$. Without it the test can fail. The function below is $f(x) = x$ for $x \le 2$ and $f(x) = 6 - x$ for $x > 2$. Its derivative changes from positive to negative at $2$, but $f$ jumps up there, and $f(2) = 2$ is smaller than the values just to its right.

On a free-response answer, name the critical point and say how $f'$ changes sign there. The scoring guides use sentences like this one:

Say what $f'$ does. A sign chart by itself doesn't count as a justification, and neither does a description of the graph of $f$.

The test needs $f'$ on both sides of $c$, so it classifies interior points only. That matches the definition: a relative extremum needs an open interval around $c$, so an endpoint of the domain is never one. AP questions about relative extrema usually give an open interval, which sidesteps the endpoint issue. An endpoint can still be where $f$ is largest or smallest on the whole interval. Comparing values to find that comes later in this unit.

To use the test on a formula, find $f'$ and factor it. A factor that's never negative, such as an even power, an exponential, or $1 - \cos x$, can't change the sign of $f'$. The sign comes from the other factors.

When the graph of $f'$ is given, read its sign: above the axis or below it. The highest and lowest points of the graph of $f'$ say nothing about the extrema of $f$.