Critical Points and the First Derivative Test
Find critical points and apply the First Derivative Test to classify each as a local max, local min, or neither. Analyze the sign of f' on each interval using a sign chart and connect to function graphs.
A lesson of AP Calculus AB, in Analytical Applications of Differentiation, about 10 minutes.
By the end of this lesson, you will be able to find all critical points of a function and use the first derivative test to classify local extrema.
Critical Points
A critical point of $f$ occurs at $x = c$ when either:
- $f'(c) = 0$, or
- $f'(c)$ does not exist (but $f(c)$ is defined)
Critical points are candidates for local maxima or minima. By Fermat's Theorem, every local extremum must occur at a critical point.
The First Derivative Test
At a critical point $c$:
| $f'$ behavior around $c$ | Conclusion |
|---|---|
| Changes from + to − (left to right) | Local maximum at $c$ |
| Changes from − to + (left to right) | Local minimum at $c$ |
| Doesn't change sign | Neither (saddle point) |
Process:
- Find $f'(x)$
- Find critical points: $f'(c) = 0$ or $f'(c)$ undefined
- Make a sign chart for $f'$
- Identify sign changes → classify extrema
Example
$f(x) = x^3 - 6x^2 + 9x + 1$
$f'(x) = 3x^2 - 12x + 9 = 3(x^2-4x+3) = 3(x-1)(x-3)$
Critical points: $x = 1$ and $x = 3$
Sign chart for $f'$:
| Interval | Test point | $(x-1)$ | $(x-3)$ | $f'$ | $f$ |
|---|---|---|---|---|---|
| $x < 1$ | $x=0$ | − | − | + | Increasing |
| $1 < x < 3$ | $x=2$ | + | − | − | Decreasing |
| $x > 3$ | $x=4$ | + | + | + | Increasing |
Conclusion:
- At $x = 1$: $f'$ changes from + to − → local maximum. $f(1) = 1-6+9+1 = 5$.