When Is a Graph Concave Down? How to Tell in 2025

A graph is concave down on any interval where its second derivative is negative (f″(x) < 0). That makes the curve bend downward like a frown, and its slopes are decreasing as you move left to right. This is the direct test: if the second derivative is less than zero across an interval, the graph is concave down there. Knowing when a graph is concave down matters because the shape tells you how a quantity’s rate of change is itself changing — a core idea in calculus, physics, and economics.

What Concave Down Means

Concavity describes how a curve bends, not whether it rises or falls. A function is concave down when its graph bows downward on an interval — the tangent lines sit above the curve, and the shape resembles a hilltop or a frowning mouth (∩). A function is concave up when the graph bows upward — tangent lines sit below the curve, and the shape looks like a valley or a smile (∪).

The word “concave” alone just means “curving inward,” so always ask: inward relative to what? Concave down curves inward toward the bottom; concave up curves inward toward the top. The Khan Academy treatment of concavity frames it the same way — as a description of the turning of the slope.

Picture a simple arch, like the top half of a circle. Near the peak, the curve flattens out exactly at the top. Move away from the peak in either direction and the curve steepens as it drops. That flatten-then-steepen behavior is the fingerprint of a downward-opening curve, and it appears on any concave-down interval.

Crucially, a graph can switch concavity. One interval might be concave up, the next concave down. The point where the switch happens is called an inflection point.

The Concavity Test

Concavity is measured with the second derivative, written f″(x). It is the derivative of the first derivative f′(x), so it captures how the slope itself is changing. Paul’s Online Math Notes states the rule cleanly: where f″(x) < 0, the function is concave down; where f″(x) > 0, it is concave up.

A Worked Example

Take f(x) = −x² + 4x. Its first derivative is f′(x) = −2x + 4, and its second derivative is f″(x) = −2.

Since −2 is negative for every value of x, this parabola is concave down everywhere. That matches the picture: y = −x² + 4x opens downward, with its single peak where f′(x) = 0, i.e. x = 2. There is no inflection point because the sign of f″ never flips.

Now consider f(x) = x³ − 3x. Here f′(x) = 3x² − 3 and f″(x) = 6x. The sign of 6x depends on x:

• When x < 0, f″(x) < 0 → concave down.
• When x = 0, f″(x) = 0 → possible inflection point.
• When x > 0, f″(x) > 0 → concave up.

Because the concavity actually changes at x = 0, that point is a genuine inflection point. MIT OpenCourseWare notes that f″(x) = 0 is only a candidate for an inflection point — the sign must change to confirm it.

Quick Reference Table

Reading Your Graph

You do not always need calculus to spot a concave-down region. Visually, look for the parts of a curve that arch over like a dome or the rim of a cup you are looking into. The peak of a hill, the top of a bouncing ball’s path mid-flight, and the upper arc of a rainbow are all concave down.

The deeper signal is in the slopes. On a concave-down interval, the slope decreases as you move from left to right. That decrease can happen in several ways:

• The curve rises but is flattening out (slope shrinking toward zero).
• The curve is level at a peak (slope near zero).
• The curve is falling and getting steeper (slope going more negative).

All three are the same underlying fact: f′ is decreasing, so f″ is negative. This is the trick that links the visual “frown” to the algebraic test. If you can trace a curve with your finger and feel the tangent tilting steadily downward as you slide right, you are on a concave-down stretch.

A useful check: mark the flat spot at the top of any hump. Just left of it the slope is positive; just right of it the slope is negative. A slope that goes from positive to negative has decreased, confirming concave down near the peak.

Key Mistakes to Avoid

The most common error is treating concavity as a statement about whether the function is rising or falling. It is not. Increasing/decreasing is governed by the first derivative f′(x); concavity is governed by the second derivative f″(x). A function can be concave down while increasing (rising and flattening), concave down while decreasing (falling and steepening), or concave up in either direction too. These are independent facts about a curve.

The second mistake is assuming any point where f″(x) = 0 is concave down or an inflection point. It is neither automatically. You must test the sign of f″ on both sides. For f(x) = x⁴, f″(x) = 12x², which equals zero at x = 0 — but it is positive on both sides, so the curve stays concave up and x = 0 is not an inflection point. It only becomes one when the sign genuinely flips, as in f(x) = x³ − 3x above.

A third slip is reading an inflection point as part of a concave-down region. The inflection point is the boundary between concave up and concave down; it belongs to neither interval exclusively.

Key Takeaways

• A graph is concave down where f″(x) < 0, bending downward like a frown (∩).
• Concave up (∪) means f″(x) > 0; the two describe how a curve bends, not whether it rises or falls.
• On a concave-down interval, the slope decreases as you move left to right — the tangent line tilts steadily downward.
• An inflection point is where concavity switches, marked by a sign change in f″ — not merely by f″(x) = 0.
• Increasing/decreasing comes from f′(x); concavity comes from f″(x). Do not mix them up.