Concavity & Points of Inflection
Concavity and points of inflection are controlled by the second derivative: \(f''>0\) is concave up, \(f''<0\) is concave down, and an inflection is where concavity changes.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
A point of inflection is where a curve changes concavity. This Year 12 Mathematics Advanced topic (MAV-12-06) uses \(f''\) to find concavity and inflections.
\(f''>0\) concave up; \(f''<0\) concave down. A point of inflection is where \(f''(x)=0\) and \(f''\) changes sign. \(f''=0\) alone is not enough.
Method
- Solve \(f''(x)=0\).
- Check that \(f''\) changes sign there.
- Find the \(y\)-value from \(f\).
| \(6x-6\) | \(=\) | \(0\Rightarrow x=1\) |
| \(f(1)\) | \(=\) | \(-2\) |
| \(6x-6\) | \(>\) | \(0\) |
| \(x\) | \(>\) | \(1\) |
| \(f''(0)\) | \(=\) | \(-6<0\) |
| \(f''(2)\) | \(=\) | \(6>0\) |
Sign changes, so yes.
\(f''(-1)=-6<0\), so concave down.
Common pitfalls
Frequently asked questions
What is a point of inflection?
A point where the curve changes concavity, from concave up to down or vice versa.
How do you find a point of inflection?
Solve f double prime equals zero, then check that the second derivative actually changes sign there.
Is every solution of f double prime = 0 an inflection?
No. The second derivative must change sign; if it does not, it is not an inflection.
What is concave down?
A curve bending downward like a hill, where the second derivative is negative.