Concavity Calculator is loading…
Find the second derivative \(f”(x)\). Where \(f”(x) > 0\) the graph is concave up (shaped like a cup), and where \(f”(x) < 0\) it is concave down (shaped like a cap). Points where the concavity changes are inflection points. For \(f(x) = x^3 - 3x^2\), \(f''(x) = 6x - 6\), so the graph is concave down on \((-\infty, 1)\), concave up on \((1, \infty)\), with an inflection point at \((1, -2)\).
Concave Up vs. Concave Down
Concave up ∪
- \(f”(x) > 0\)
- The slope \(f’\) is increasing
- The graph sits above its tangent lines
- Holds water like a cup
Concave down ∩
- \(f”(x) < 0\)
- The slope \(f’\) is decreasing
- The graph sits below its tangent lines
- Spills water like an upside-down cup
Concavity is about how the graph bends, not whether it goes up or down. A curve can be rising and concave down at the same time, like the top half of a hill as you climb toward the peak.
Step-by-Step Method
- Find \(f”(x)\). Take the derivative twice. The second derivative calculator shows this part in detail.
- Find the candidates. Solve \(f”(x) = 0\), and list any points where \(f”\) is undefined.
- Split the number line. The candidates cut it into intervals.
- Test each interval. Pick any number inside and check the sign of \(f”\) there.
- Name the inflection points. Keep only the candidates where the sign really changes and \(f\) is defined.
Worked Example: \(f(x) = x^4 – 6x^2\)
\(f'(x) = 4x^3 – 12x\) and \(f”(x) = 12x^2 – 12 = 12(x – 1)(x + 1)\). The candidates are \(x = -1\) and \(x = 1\).
| Interval | Test point | Sign of \(f”\) | Concavity |
|---|---|---|---|
| \((-\infty, -1)\) | \(x = -2\) | \(f”(-2) = 36 > 0\) | Up ∪ |
| \((-1, 1)\) | \(x = 0\) | \(f”(0) = -12 < 0\) | Down ∩ |
| \((1, \infty)\) | \(x = 2\) | \(f”(2) = 36 > 0\) | Up ∪ |
The sign changes at both candidates, so the inflection points are \((-1, -5)\) and \((1, -5)\).
When \(f” = 0\) Is Not an Inflection Point
Take \(f(x) = x^4\). Then \(f”(x) = 12x^2\), which is 0 at \(x = 0\). But \(12x^2\) is positive on both sides of 0, so the graph is concave up everywhere and there is no inflection point. Always check that the sign actually changes.
The opposite can also happen: \(f”\) can be undefined at an inflection point. For \(f(x) = \sqrt[3]{x}\), the second derivative does not exist at \(x = 0\), yet the graph switches from concave up to concave down there.
Concavity and the Second Derivative Test
Concavity also tells you about peaks and valleys. If \(f'(c) = 0\) and the graph is concave up at \(c\), then \(c\) is a local minimum (the bottom of a cup). If it is concave down, \(c\) is a local maximum. Use the derivative calculator to find where \(f'(c) = 0\), then check concavity here.
Learn the Math Behind the Concavity Calculator
Concavity comes from the second derivative. Where it changes sign, the graph has an inflection point.
Frequently Asked Questions
What does the x-range box do?
The calculator scans the interval you give it (from −10 to 10 by default) for places where \(f”\) is zero or undefined. For periodic functions like \(\sin x\), use a smaller range such as 0, 2pi to see one full cycle.
Is an inflection point the same as a critical point?
No. Critical points come from \(f'(x) = 0\) and are about the slope. Inflection points come from \(f”\) changing sign and are about the bend.
Can a straight line be concave?
No. A line has \(f”(x) = 0\) everywhere, so it is neither concave up nor concave down.
What does concavity mean in real life?
If \(f\) is position, \(f”\) is acceleration. Concave up means the velocity is increasing. If \(f\) is a company’s sales, an inflection point is where growth stops speeding up and starts slowing down.