How to Determine the Concavity of a Function in 5 Clear Steps

To determine the concavity of a function, find its second derivative f’‘(x), then check its sign on each interval between candidate points: where f’‘(x) > 0 the curve is concave up (holds water), and where f’‘(x) < 0 it is concave down (spills water). Points where the second derivative equals zero or is undefined are candidates for inflection points, where concavity may switch. A sign chart of f’'(x) turns these checks into a quick, reliable routine.

This approach appears throughout standard calculus: concavity describes how a curve bends, not just whether it rises or falls.

What Concavity Means

Concavity describes the direction a curve bends. A function is concave up on an interval when its graph bends upward like a cup or a smile: the slopes of the tangent lines increase as x increases. A function is concave down when the graph bends downward like a frown, and the tangent slopes decrease.

The water test makes it concrete:

• Concave up: the curve holds water (opens upward).
• Concave down: the curve spills water (opens downward).

Concavity is always interval-based. A single function can be concave up in one region and concave down in another. Note that concavity and increasing/decreasing are separate ideas: a curve can be rising while bending downward, or falling while bending upward. For a visual treatment, Khan Academy walks through graphs of both cases.

Find the Second Derivative

The second derivative f’'(x) is the key tool for testing concavity, because it measures the rate at which the first derivative (the slope) changes. Computing it is a two-step differentiation of the original function.

Step 1: Differentiate f(x) once to get f’(x). Step 2: Differentiate f’(x) to get f’'(x).

Use the same rules you already know — power, product, quotient, and chain — applied twice. Three quick illustrations:

• f(x) = x³ → f’(x) = 3x² → f’'(x) = 6x
• f(x) = x⁴ → f’(x) = 4x³ → f’'(x) = 12x²
• f(x) = sin x → f’(x) = cos x → f’'(x) = −sin x

A Worked Example

Take f(x) = x³ − 3x² + 2.

First derivative: f’(x) = 3x² − 6x. Second derivative: f’'(x) = 6x − 6.

The sign of f’'(x) = 6x − 6 depends on whether x is greater or less than 1, which sets up the next steps.

Locate Inflection Points

An inflection point is a point on the curve where concavity actually changes — from up to down, or down to up. To locate the candidates, solve f’‘(x) = 0 and also find any x where f’'(x) is undefined.

For f’'(x) = 6x − 6: setting 6x − 6 = 0 gives x = 1. That is the only candidate.

Be careful: a zero or undefined second derivative does not guarantee an inflection point exists there. It only means concavity might change. Two common false alarms:

• f(x) = x⁴ has f’'(x) = 12x², which equals 0 at x = 0. But 12x² is never negative, so the curve is concave up on both sides — no change, no inflection point.
• For a function like f(x) = 1/x, f’'(x) = 2/x³ is undefined at x = 0, but x = 0 is not in the domain at all, so it cannot be an inflection point.

This is why every candidate must survive the sign test. Wolfram MathWorld defines an inflection point precisely as a point where the sign of the second derivative changes, underscoring that the sign change — not the zero itself — is what matters.

Test Intervals on a Sign Chart

The sign chart is where concavity gets decided. Mark every candidate point on a number line; these points divide the domain into intervals. Then pick one test value inside each interval, plug it into f’'(x), and record only the sign — positive means concave up, negative means concave down.

Continuing the example f’'(x) = 6x − 6, with candidate x = 1:

Because the sign of f’'(x) flips from negative to positive at x = 1, the curve changes from concave down to concave up there — confirming a genuine inflection point at x = 1. The function f(x) = x³ − 3x² + 2 is concave down on (−∞, 1) and concave up on (1, ∞).

A few practical notes:

• Any test number in the interval works; the sign in the middle holds throughout, provided f’’ is continuous and has no other zeros there.
• A sign change at a candidate confirms an inflection point. No sign change means no inflection point, as with x⁴ at x = 0.
• Write the final answer as intervals — for example, “concave up on (1, ∞)” — since concavity is a property of whole intervals, not single points.

Common Mistakes to Avoid

Even students who know the method slip in predictable ways. Watch for these traps:

• Confusing concavity with direction. A function can be decreasing and concave up at the same time. The two classifications use different derivatives and answer different questions.
• Treating every zero of f’'(x) as an inflection point. Always verify with a sign change, as the x⁴ case shows.
• Skipping undefined points. Rational and radical functions often have f’'(x) undefined at values inside the domain; those belong on the sign chart too.
• Plugging the candidate back in instead of a test value. You need a value strictly inside each interval, not the boundary point.
• Forgetting the domain. Points outside the domain (like x = 0 for 1/x) can never be inflection points, no matter what the second derivative does.

A quick sanity check: sketch the curve roughly and ask whether your intervals match what you see. If the algebra says concave up but the sketch bends the other way, recheck the second derivative — an arithmetic slip in differentiation is the usual culprit.

When the Second Derivative Is Hard to Compute

Some functions make f’'(x) messy, but the method does not change. For a product like f(x) = x·eˣ, apply the product rule twice:

• f’(x) = eˣ + x·eˣ = eˣ(1 + x)
• f’'(x) = eˣ(1 + x) + eˣ = eˣ(2 + x)

Since eˣ is always positive, the sign of f’'(x) matches the sign of (2 + x). So the curve is concave down for x < −2 and concave up for x > −2, with an inflection point at x = −2. Recognizing that a positive factor can be ignored for sign purposes saves time and reduces errors.

For implicit curves and more advanced settings, references such as Wolfram MathWorld document the underlying definitions of second derivatives and points of inflection, which remain the foundation whatever the function’s complexity.

Key Takeaways

• Concavity is the direction a curve bends: concave up (opens like a cup, f’‘(x) > 0) or concave down (opens like a frown, f’'(x) < 0).
• The second derivative f’'(x) is the test tool — compute it by differentiating twice.
• Candidate inflection points come from f’‘(x) = 0 or f’'(x) undefined; candidates are not automatically inflection points.
• A sign chart of f’'(x) across intervals classifies concavity and confirms inflection points only where the sign actually changes.
• State your answer as intervals, and remember that concavity is separate from whether a function is increasing or decreasing.