Concavity?

Whether a curve bends upward like a bowl or downward like a hill — determined by the second derivative, and key to classifying critical points.

Simple Definition

Concavity describes the direction a curve bends. A curve is concave up if it bends upward — like the inside of a bowl. It's concave down if it bends downward — like the top of a hill.

The second derivative \(f''(x)\) tells you which: if \(f''(x) > 0\), the curve is concave up; if \(f''(x) < 0\), it's concave down. When the concavity switches — from up to down or down to up — that's an inflection point.

Concavity is the missing piece that completes curve sketching. The first derivative tells you where the function increases and decreases. The second derivative tells you how those changes are accelerating or decelerating.

Concave Up vs Concave Down

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Concave Up

\(f''(x) > 0\)

Curve bends upward like a bowl. Tangent lines lie below the curve. The slope is increasing. Think: "holds water."

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Concave Down

\(f''(x) < 0\)

Curve bends downward like a hill. Tangent lines lie above the curve. The slope is decreasing. Think: "spills water."

Intuition

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Speed of the slope

The first derivative is the slope (speed). The second derivative is the rate of change of the slope (acceleration). Concave up means the slope is speeding up — the curve is accelerating upward. Concave down means the slope is slowing down — it's decelerating. This is exactly the relationship between position, velocity, and acceleration in physics.

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Where do tangent lines go?

For a concave-up curve, every tangent line lies below the curve — the curve curves away from the tangent upward. For a concave-down curve, every tangent line lies above — the curve falls away below the tangent. This geometric test always works, and it's why concavity matters for the second derivative test for extrema.

Visual — Concavity & Inflection Points

Green region = concave up (f″ > 0) · Red region = concave down (f″ < 0) · Gold dot = inflection point

Key Rules & the Second Derivative Test

Concave up	\(f''(x) > 0\) on an interval
Concave down	\(f''(x) < 0\) on an interval
Inflection point	\(f''(x) = 0\) AND concavity actually changes at \(x\)
2nd deriv test: local min	\(f'(c) = 0\) and \(f''(c) > 0\) → local minimum
2nd deriv test: local max	\(f'(c) = 0\) and \(f''(c) < 0\) → local maximum
2nd deriv test: inconclusive	\(f'(c) = 0\) and \(f''(c) = 0\) → use 1st deriv test

Quick Examples

Example 1 — Full analysis

\(f(x) = x^3 - 3x\)

\(f''(x) = 6x\)
Concave up: \(x > 0\)
Concave down: \(x < 0\)
Inflection at \(x = 0\)

Example 2 — Second deriv test

\(f(x) = x^4 - 4x^2\)

\(f'(x) = 4x^3-8x=0 \Rightarrow x=0,\pm\sqrt{2}\)
\(f''(0) = -8 < 0\) → local max
\(f''(\pm\sqrt{2}) = 16 > 0\) → local min

Example 3 — f″ = 0 but no inflection

\(f(x) = x^4\)

\(f''(x) = 12x^2\)
\(f''(0) = 0\) but \(f''\) doesn't change sign
→ No inflection at \(x = 0\)

Example 4 — Parabola

\(f(x) = x^2\)

\(f''(x) = 2 > 0\) everywhere
Concave up on all of ℝ
No inflection points

Common Mistakes

✗ "f″(c) = 0 means c is an inflection point"

Not necessarily. \(f''(c) = 0\) is a necessary condition for an inflection point, but not sufficient. You must also verify that \(f''\) changes sign at \(c\). For \(f(x) = x^4\), \(f''(0) = 0\) but concavity doesn't change — so there's no inflection point.

✗ Confusing concave up with increasing

A function can be decreasing and concave up at the same time. "Concave up" means the slope is increasing — but the slope could be increasing from a large negative value toward zero. The function could still be going down while bending upward.

✗ Applying the 2nd deriv test when f″(c) = 0

When \(f''(c) = 0\), the second derivative test is completely inconclusive. You must fall back on the first derivative test — check the sign of \(f'\) on both sides of \(c\) to classify the critical point.

Related Concepts

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