Unit 5: Analytical Applications of Differentiation

The Second Derivative Test & Concavity

Using the second derivative to determine concavity and classify extrema.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The First Derivative Test

What the Derivative of the Derivative Reveals

The first derivative test requires checking a sign on both sides of a critical point. Before reading on: if f'' — the rate of change of the slope itself — is positive at a critical point, what would that say about which way the curve is bending there, and what kind of extremum that implies?

Definition — Concavity and the Second Derivative Test

f is concave up where f''(x) > 0 (the graph curves upward, like a cup) and concave down where f''(x) < 0 (curves downward, like a frown). A point of inflection is where concavity actually changes. At a critical point c where f'(c) = 0: if f''(c) > 0, c is a local minimum; if f''(c) < 0, c is a local maximum; if f''(c) = 0, the test is inconclusive and the first derivative test is needed instead.

Worked Example — Confirming Extrema with the Second Derivative Test

Using f(x) = x³ − 3x² − 9x + 5 from the previous lesson, with f'(x) = 3x² − 6x − 9 and critical points x = −1, 3: f''(x) = 6x − 6. At x = −1: f''(−1) = −6 − 6 = −12 < 0 — concave down, confirming the local maximum found by the first derivative test. At x = 3: f''(3) = 18 − 6 = 12 > 0 — concave up, confirming the local minimum. Both tests agree, as they should.

Worked Example — Finding a Point of Inflection

Find the inflection point of the same f(x) = x³ − 3x² − 9x + 5. Set f''(x) = 0: 6x − 6 = 0, so x = 1. Confirm concavity actually changes: f''(0) = −6 < 0 (concave down for x < 1), f''(2) = 6 > 0 (concave up for x > 1) — a genuine sign change, confirming a real inflection point. f(1) = 1 − 3 − 9 + 5 = −6, so the inflection point is (1, −6).

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

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

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
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

Finding where f''(x) = 0 only locates a candidate inflection point — always confirm concavity actually changes sign on either side, the same caution the first derivative test uses for critical points that don't change sign.

Common Mistakes

Concluding a point is an inflection point just because f''(x) = 0 there, without checking that concavity actually changes.
f''(x) = 0 is necessary but not sufficient — a function like f(x) = x⁴ has f''(0) = 0, but concavity stays up on both sides (never changing), so x = 0 isn't an inflection point.
Using the second derivative test at a point where f''(c) = 0, and forcing a conclusion anyway.
f''(c) = 0 makes the second derivative test inconclusive by definition — fall back to the first derivative test's sign analysis at that specific point instead of guessing.

Key Takeaways

f''(x) > 0 means concave up; f''(x) < 0 means concave down; a point of inflection is where concavity genuinely changes.
At a critical point, the second derivative test classifies it by the sign of f'' there — positive for a local min, negative for a local max — often faster than a full sign analysis of f'.
The second derivative test is inconclusive when f''(c) = 0, requiring the first derivative test as a fallback.

Summary

Extrema and concavity fully describe a function's local shape. The next lesson applies exactly this machinery to a genuinely applied problem — finding the best possible value of some real quantity.

Track your progress

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