Applications of Derivatives
The First Derivative Test
Using the first derivative to find critical points, local extrema, and intervals of increase/decrease.
Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed
Where a Graph Turns, According to Its Slope
Definition — Critical Point and the First Derivative Test
A critical point of f is a value c where f'(c) = 0 or is undefined. The first derivative test classifies it: if f' changes from positive to negative at c, f has a local maximum there; if negative to positive, a local minimum; if f' doesn't change sign, neither.
Worked Example — Classifying Critical Points with a Sign Analysis
Find and classify the critical points of f(x) = x³ − 3x² − 24x + 10. f'(x) = 3x² − 6x − 24 = 3(x² − 2x − 8) = 3(x−4)(x+2). Critical points: x = −2, 4. Test each interval: f'(−3) = 3(−7)(−1) = 21 > 0 (increasing); f'(0) = 3(−4)(2) = −24 < 0 (decreasing); f'(5) = 3(1)(7) = 21 > 0 (increasing). f' changes + to − at x = −2: local max, f(−2) = −8 − 12 + 48 + 10 = 38. f' changes − to + at x = 4: local min, f(4) = 64 − 48 − 96 + 10 = −70.
Derivative Explorer
Point of tangency
1
f(x), with tangent line at x = a
Derivative graph — linked to the graph above; panning or zooming either moves both
What's happening at x = a
- At x = 1, f(x) ≈ -2 — the point (1, -2).
- The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
- That slope is positive, so f is increasing at this point.
- The second derivative is positive here, so f is concave up (curving upward) near this point.
- Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.
Numerical derivative comparison
| h | Forward | Backward | Central |
|---|---|---|---|
| 1 | 4 | -2 | 1 |
| 0.1 | 0.31 | -0.29 | 0.01 |
| 0.01 | 0.0301 | -0.0299 | 0.0001 |
| 0.001 | 0.003 | -0.003 | 0 |
Analysis of f(x)
- y-intercept
- (0, 0)
- x-intercepts
- (-1.73, 0), (0, 0), (1.73, 0)
- Extrema
- local max at (-1, 2); local min at (1, -2)
- Inflection points
- (0, 0)
- Vertical asymptotes
- none found in view
- Horizontal asymptotes
- none found
- Domain
- all real numbers in view
- Range (estimated)
- approximately [-970, 970]
Analysis of f'(x) — its roots are f's critical points
- y-intercept
- (0, -3)
- x-intercepts
- (-1, 0), (1, 0)
- Extrema
- local min at (0, -3)
- Inflection points
- none found in view
- Vertical asymptotes
- none found in view
- Horizontal asymptotes
- none found
- Domain
- all real numbers in view
- Range (estimated)
- approximately [-3, 297]
Tip
A sign analysis needs only one test point per interval between consecutive critical points — f' can't change sign without passing through zero or an undefined point.
Common Mistakes
Assuming every solution of f'(c) = 0 is automatically a local extremum.
f'(c) = 0 only makes c a critical point, a candidate — the sign of f' must actually change on either side to confirm a max or min.
Key Takeaways
- A critical point occurs where f'(c) = 0 or is undefined, and is only a candidate for a local extremum.
- The first derivative test classifies a critical point by whether f' changes sign, and how, around it.
Summary
The next lesson reads a second layer of shape — concavity — from the second derivative, and offers a faster classification test.
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