Concavity describes how the slope of a curve changes. A function is concave up where its slopes are increasing and concave down where its slopes are decreasing. For a twice-differentiable function, the second derivative provides the standard test:
- \(f”(x)>0\): concave up;
- \(f”(x)<0\): concave down.
1. Work through a cubic example
Let
\[f(x)=x^3-3x.\]
Then
\[f'(x)=3x^2-3,\qquad f”(x)=6x.\]
The second derivative is negative for \(x<0\) and positive for \(x>0\). Therefore the graph is concave down on \(( -\infty,0)\) and concave up on \((0,\infty)\).
2. Identify the inflection point
A candidate occurs where \(f”(x)=0\) or where \(f”\) is undefined. Here \(6x=0\) gives \(x=0\). Because the sign of \(f”\) changes, \(x=0\) is an inflection x-value. Since \(f(0)=0\), the inflection point is \((0,0)\).
3. Why f″ = 0 is not enough
For \(g(x)=x^4\),
\[g”(x)=12x^2.\]
Although \(g”(0)=0\), the second derivative is nonnegative on both sides and the graph remains concave up. Therefore \((0,0)\) is not an inflection point.
4. An inflection point can occur where f″ is undefined
Consider \(h(x)=x^{1/3}\). Its second derivative is undefined at zero, but the graph changes concavity there and the function is continuous. Thus \((0,0)\) is an inflection point. The definition depends on a change in concavity, not merely on solving \(f”(x)=0\).
5. Build a second-derivative sign chart
- Find the domain of f.
- Compute \(f”(x)\).
- List zeros and undefined points of \(f”\) that lie in or border the domain.
- Test the sign on every interval.
- Check whether concavity changes at each candidate.
- Evaluate f to obtain point coordinates.
The Variation Table tool includes a first-and-second-derivative analysis format that can summarize these intervals alongside increasing and decreasing behavior.
6. Connect concavity with tangent lines
On a concave-up interval, the graph typically lies above its tangent lines near the point of tangency. On a concave-down interval, it typically lies below them. This geometric interpretation helps detect errors in a second-derivative table.
Common mistakes
- Calling every solution of \(f”(x)=0\) an inflection point.
- Forgetting points where \(f”\) is undefined.
- Reporting an x-value without the y-coordinate when a point is requested.
- Ignoring continuity at a proposed inflection point.
- Confusing increasing/decreasing with concave up/down.
Graph the cubic example in the Graphing Calculator and compare it with first- and second-derivative analysis.