Applications of Derivatives
The Second Derivative & Concavity
Using the second derivative to determine concavity and inflection points.
Advanced20 min lesson1 min readUpdated August 12, 2026Author not yet attributed
Prerequisites
- The First Derivative Test
What the Derivative of the Derivative Reveals
Definition — Concavity and the Second Derivative Test
f is concave up where f''(x) > 0 and concave down where f''(x) < 0. A point of inflection is where concavity genuinely changes. At a critical point where f'(c) = 0: f''(c) > 0 means a local minimum, f''(c) < 0 means a local maximum, and f''(c) = 0 is inconclusive.
Worked Example — Confirming Extrema and Finding an Inflection Point
Using f(x) = x³ − 3x² − 24x + 10 from the previous lesson, with f'(x) = 3x² − 6x − 24 and critical points x = −2, 4: f''(x) = 6x − 6. At x = −2: f''(−2) = −18 < 0 — concave down, confirming the local maximum. At x = 4: f''(4) = 18 > 0 — concave up, confirming the local minimum. Setting f''(x) = 0: x = 1. Checking f''(0) = −6 < 0 and f''(2) = 6 > 0 confirms a genuine sign change, so x = 1 is a real inflection point: f(1) = 1 − 3 − 24 + 10 = −16.
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
f''(x) = 0 only locates a candidate inflection point — always confirm concavity actually changes sign on either side.
Common Mistakes
Concluding an inflection point exists just because f''(x) = 0, without checking that concavity changes.
f''(x) = 0 is necessary but not sufficient — concavity has to genuinely switch sign on either side to confirm a real inflection point.
Key Takeaways
- f''(x) > 0 means concave up; f''(x) < 0 means concave down; an inflection point is where concavity genuinely changes.
- At a critical point, the sign of f'' offers a faster alternative to a full first-derivative sign analysis.
Summary
This closes curve sketching. The next lesson applies exactly this machinery to genuinely applied problems, starting with finding the best possible value of a real quantity.
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