First Derivative Test
The first derivative test classifies a critical point as a local maximum, local minimum, or neither by checking whether the derivative changes sign there.
Definition
After finding a critical point, the first derivative test checks the sign of the derivative just before and just after that point: if the sign goes from $+$ to $-$, it is a local max, if it goes from $-$ to $+$, it is a local min, and if the sign does not change, the point is neither (possibly an inflection point). This sign-analysis technique requires only the sign of $f'$ near $c$, not its magnitude, so it is applicable even when $f''$ does not exist, and it follows directly from the relationship between monotonicity and the sign of $f'$ established by the Mean Value Theorem.
Example
Think of hiking: if you were going downhill and then start going uphill, you just passed through a valley (local min); downhill then uphill means you hit a peak (local max). For $f(x) = x^3 - 3x$, critical points at $x = \pm 1$: at $x = -1$, $f'$ goes from $+$ to $-$ (local max, $f(-1)=2$); at $x = 1$, $f'$ goes from $-$ to $+$ (local min, $f(1)=-2$). Even where the derivative does not exist, sign analysis still works: $f(x) = x^{2/3}$ has a local min at $0$ even though $f'(0)$ is undefined.
Key Insight
The first derivative test is more versatile than the second derivative test because it works even when $f'' = 0$ at the critical point or does not exist there. In variational calculus, the analogue of the first derivative test is the Euler-Lagrange equation: a functional has a critical point where its "derivative" (functional derivative) vanishes.