Concavity Calculator

The concavity calculator determines where a function is concave up or concave down by analysing the sign of the second derivative across its domain. It is used by calculus students studying curve behaviour, economists interpreting acceleration in trends, and anyone who needs to understand how the shape of a curve changes. Enter your function to get concavity intervals and any inflection points.

How to Use the Concavity Calculator

1. Enter your function f(x) in the Concavity Calculator input field. Click Calculate to find f''(x) and solve f''(x) = 0.
2. Review the sign chart for f''(x) across the number line.
3. Read the concavity intervals: where f''(x) > 0 the function is concave up; where f''(x) < 0 it is concave down.
4. Note any inflection points listed where f'' changes sign.

The Formula

Concavity is determined by the second derivative f''(x):

• Concave up on an interval if f''(x) > 0 for all x in that interval (the curve opens upward like a bowl)
• Concave down on an interval if f''(x) < 0 for all x in that interval (the curve opens downward like an arch)
• Inflection point at x = c if f''(c) = 0 and f'' changes sign at c

To analyse concavity:

1. Find f''(x)
2. Solve f''(x) = 0 to find candidate inflection points
3. Test the sign of f''(x) in each interval created by these points

Real-World Example

Determine the concavity of f(x) = x^3 - 3x^2 + 2.

1. Find f'(x) = 3x^2 - 6x
2. Find f''(x) = 6x - 6
3. Set f''(x) = 0: 6x - 6 = 0, so x = 1
4. Test intervals:
   • For x < 1: pick x = 0, f''(0) = -6 < 0, so concave down on (-infinity, 1)
   • For x > 1: pick x = 2, f''(2) = 6 > 0, so concave up on (1, infinity)
5. Since f'' changes sign at x = 1, the function has an inflection point at (1, f(1)) = (1, 0)

The function is concave down to the left of x = 1 and concave up to the right.

Why Concavity Matters

Concavity gives you more information about a function than the first derivative alone. Knowing a function is increasing tells you it is going up, but knowing it is concave down while increasing tells you the growth is slowing. This distinction is important in economics: a revenue curve that is increasing but concave down indicates diminishing returns. In biology, a population growing but with decreasing concavity is approaching a carrying capacity. The second derivative provides this second layer of insight, converting a bare rate of change into a richer picture of how the rate itself is evolving.

Frequently Asked Questions

What is the difference between concave up and convex? These terms describe the same shape from opposite perspectives. Concave up (or convex downward) means the curve opens like a bowl with the opening facing upward. Some textbooks use "convex" to mean concave down; always check the convention used in your course material to avoid confusion.

How does concavity relate to the second derivative test for critical points? When you find a critical point where f'(c) = 0, the second derivative test uses concavity to classify it. If f''(c) > 0 (concave up), the critical point is a local minimum. If f''(c) < 0 (concave down), it is a local maximum.

Can a function be both concave up and concave down at the same point? No. At any given point, a function is either concave up, concave down, or at an inflection point where the concavity changes. A single point cannot have two concavity states simultaneously.

Does a straight line have concavity? A straight line has f''(x) = 0 everywhere, so it is neither concave up nor concave down. It is considered linear rather than curved, and it has no inflection points.

Also try these free tools:

• Inflection Points Calculator
• Critical Points Calculator
• Derivative Calculator

A quartic worked end to end

Take f(x) = x^4 - 4x^2. Differentiate once for f'(x) = 4x^3 - 8x, then again for f''(x) = 12x^2 - 8.

Setting f''(x) = 0 gives 12x^2 = 8, so x^2 = 8/12 = 0.666667 and x = plus or minus 0.816497. Those are the only candidate inflection points, and they cut the number line into three intervals.

At x = 0.816497 the function value is 0.816497^4 minus 4 times 0.816497^2, which is 0.444444 minus 2.666667, or minus 2.222222. The same value appears at the negative root because the function is even, so the two inflection points are (0.816497, minus 2.222222) and (minus 0.816497, minus 2.222222). In exact fractions the ordinate is minus 20/9.

The first derivative has roots at x = 0 and at x = plus or minus 1.414214, so there are three critical points. The second derivative at x = 0 is minus 8, and a negative second derivative at a critical point means a local maximum, so this quartic peaks at the origin, falls away on both sides, and turns upward once it passes the inflection points.

Sign chart for the second derivative

One test value inside each interval is enough once the roots are known. The table lists f''(x) = 12x^2 - 8 across the interval, with the concavity each sign implies.

The sign is positive outside the two roots and negative between them, so the curve is concave up on the two outer intervals and concave down between minus 0.816497 and 0.816497. Each sign change lines up with a root, which is what makes both roots genuine inflection points.

Checking a candidate point before you call it an inflection

A root of f''(x) is a candidate, not a conclusion. The quartic above earns the name because the sign changes across each root. The function f(x) = x^4 shows what happens when it does not.

For f(x) = x^4 the second derivative is 12x^2, which is zero at x = 0 and positive everywhere else. The sign does not change at the root, so the curve is concave up on both sides of the origin and there is no inflection point there. A page that marked every zero of f'' as an inflection would report one for x^4, and the report would be wrong.

The rule for a reader checking work by hand: find the roots of f'', test one point on each side of each root, and claim an inflection only where the sign flips. Three test points settle a quadratic second derivative, and a cubic second derivative may need four.

The third derivative as a shortcut

The sign test always works, and there is a faster check when the third derivative is easy to take. For the quartic above, f'''(x) = 24x. At x = 0.816497 the third derivative is 19.595918, and at x = minus 0.816497 it is minus 19.595918.

A non-zero third derivative at a point where f'' is zero means f'' is changing at that point, so its sign flips and the point is a genuine inflection. No interval testing is needed. The two routes agree here and they agree in general whenever the third derivative exists and does not vanish.

Where f''' is also zero, the shortcut fails and the sign chart is the only route. That case is the stationary inflection, where f'' touches zero without crossing, and the x^4 example above is the simplest member of the family.

Slope and curvature together

The first derivative describes the direction of the curve and the second describes its bend. Reading the two columns side by side is what shows why the quartic looks the way it does.

At x = 0 the slope is zero and the curvature is negative, so the curve tops out there. Between x = 0 and x = 1 the slope turns from zero to minus 4 while the curvature is positive, so the curve is already bending back upward even as it falls. That combination, a falling slope with concave up curvature, produces the minimum at x = 1.414214 one interval after the curve crosses its inflection point at x = 0.816497.

Method, assumptions and where the test stops working

Four conditions sit under the interval analysis, and each one fails on some functions.

• The second derivative must exist on the interval. A function with a corner or a vertical tangent, such as the absolute value at the origin, has no second derivative there and the test does not apply.
• The function must be twice differentiable on the open interval, not merely once. Concavity is a statement about f'', so a point where f'' is undefined needs separate handling.
• Concavity at the exact point of inflection is neither up nor down. The intervals are open at any endpoint that is a root of f''.
• The sign chart only carries meaning inside the domain. Find the domain first and the chart second, or the answer will describe points the function never reaches.

Where the conditions hold, the arithmetic is short: two derivatives, one quadratic to solve, and a sign test per interval. Where they fail, the sign chart gives a confident answer to the wrong question.

Reading the result on a graph

Concave up means the curve sits above its tangent line and the slopes increase from left to right. Concave down means the curve sits below its tangent and the slopes decrease. At an inflection point the tangent line crosses the curve rather than touching it from one side.

Those three sentences let you check the algebra against a sketch. If the algebra says the curve is concave down between the two roots and your sketch shows a bowl over that stretch, one of the two is wrong, and the sign chart shows which.