Unit 5: Analytical Applications of Differentiation
First & Second Derivative Tests
Using derivative tests to find extrema and determine concavity.
Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed
Reading a Graph's Shape from Its First and Second Derivatives
Definition — Critical Points and the Derivative Tests
A critical point is where f'(c) = 0 or is undefined. The first derivative test classifies it by whether f' changes from + to − (local max), − to + (local min), or doesn't change (neither). The second derivative test offers a shortcut at a critical point where f'(c) = 0: f''(c) > 0 means a local min, f''(c) < 0 means a local max, and f''(c) = 0 is inconclusive. f''(x) > 0 means concave up; f''(x) < 0 means concave down; a point of inflection is where concavity genuinely changes.
Worked Example — A Complete Analysis: Extrema, Concavity, and Inflection
Analyze f(x) = 2x³ − 3x² − 12x + 5. f'(x) = 6x² − 6x − 12 = 6(x−2)(x+1), critical points x = −1, 2. Sign test: f'(−2) = 24 > 0, f'(0) = −12 < 0, f'(3) = 24 > 0 — increasing, then decreasing, then increasing. f' changes + to − at x = −1 (local max) and − to + at x = 2 (local min). f(−1) = −2−3+12+5 = 12; f(2) = 16−12−24+5 = −15. Confirm with f''(x) = 12x − 6: f''(−1) = −18 < 0 (concave down, confirms local max); f''(2) = 18 > 0 (concave up, confirms local min) — both tests agree. Set f''(x) = 0: x = 1/2. Since f''(0) = −6 < 0 and f''(1) = 6 > 0, concavity genuinely changes there: f(1/2) = 0.25−0.75−6+5 = −1.5, an inflection point at (0.5, −1.5).
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 critical point isn't automatically an extremum — f(x) = x³ has f'(0) = 0, but f'(x) = 3x² ≥ 0 everywhere, never changing sign, so x = 0 is neither a max nor a min (it's an inflection point instead, since f''(x) = 6x does change sign there).
Common Mistakes
Using the second derivative test where f''(c) = 0.
That case is inconclusive by definition — fall back to the first derivative test's sign analysis at that specific point.
Key Takeaways
- A critical point is a candidate extremum, classified by how f' changes sign around it, or by the sign of f'' there.
- f''(x) determines concavity; a point of inflection is where concavity genuinely changes, confirmed by a sign change in f''.
- Both derivative tests, applied to the same function, must agree — a useful cross-check.
Summary
The next lesson applies exactly this machinery to a genuinely applied problem — finding the best possible value of a real quantity.
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