Concavity and points of inflection
Concavity and Points of Inflection
Concavity and Points of Inflection is a topic in Applications of Differentiation in the California Calculus Standards. It is aligned to Standard 9.0, which requires students to use differentiation to sketch graphs by hand, here using the second derivative for concavity and points of inflection.
Concavity describes how a curve bends: it is concave up where \(f''(x)>0\) and concave down where \(f''(x)<0\). An inflection point is where the concavity changes.
Theory
Concavity describes how a curve bends. Where \(f''>0\) the graph is concave up (opens upward); where \(f''<0\) it is concave down. An inflection point is where the concavity switches.
The second derivative controls the bending of the graph:
- \(f''(x)>0\) \(\Rightarrow\) concave up (holds water);
- \(f''(x)<0\) \(\Rightarrow\) concave down (spills water).
An inflection point is where the concavity changes — \(f''\) must equal \(0\) (or be undefined) and actually switch sign there.
The concavity test:
Inflection points satisfy
How to analyze concavity
- Find \(f''(x)\) and set it equal to \(0\).
- Test the sign of \(f''\) on each interval.
- Mark inflection points where \(f''\) changes sign; \(+\) is concave up, \(-\) is concave down.
Use the sign of \(f''\).
| \(f''(x)\) | \(=\) | \(6x\) |
| \(f''<0\) | \(\Rightarrow\) | \(x<0\ \text{(concave down)}\) |
| \(f''>0\) | \(\Rightarrow\) | \(x>0\ \text{(concave up)}\) |
\(f''\) changes sign at \(x=0\): inflection at \((0,0)\).
Set \(f''=0\) and confirm a sign change.
| \(f''(x)\) | \(=\) | \(6x-6=0\) |
| \(x\) | \(=\) | \(1\) |
\(f''\) changes sign at \(x=1\); the inflection point is \((1,-2)\).
Check the sign of \(f''\) there.
| \(f''(x)\) | \(=\) | \(6x\) |
| \(f''(2)\) | \(=\) | \(12>0\) |
Since \(f''(2)>0\), the curve is concave up at \(x=2\).
\(f''(x)=12x^2\ge 0\) for all \(x\).
| \(f''(x)\) | \(=\) | \(12x^2\) |
Concave up everywhere; \(f''=0\) at \(x=0\) but there is no sign change, so no inflection point.
Common pitfalls
Frequently asked questions
What does concave up mean?
The graph opens upward like a cup, which happens where the second derivative is positive, \(f''>0\).
What is an inflection point?
A point where the concavity changes from up to down or down to up. There \(f''=0\) (or is undefined) and \(f''\) changes sign.
How do you find inflection points?
Solve \(f''(x)=0\), then check that \(f''\) actually changes sign across each solution. Only those are inflection points.
Does f''=0 always give an inflection point?
No. \(f(x)=x^4\) has \(f''(0)=0\) but is concave up on both sides, so there is no inflection there. A sign change is required.