Concavity & Points of Inflection
Learn concavity and points of inflection for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). The second derivative shows which way a curve bends, and a point of inflection is where the bending switches from one direction to the other.
You will learn to decide where a curve is concave up or concave down, solve \(f''(x) = 0\) and test either side to confirm a genuine point of inflection, and use the second derivative to classify stationary points — revealing the full shape of a graph.
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.