Points of inflection
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Points of Inflection
Find where a curve changes concavity and avoid the common exam traps.
Definition
A point of inflection is a point where the curve changes concavity: from concave up to concave down, or from concave down to concave up.
\[ f''(x) \text{ changes sign} \]
The equation \(f''(x)=0\) gives candidates, but the sign change confirms the point of inflection.
Worked example
Find the point of inflection of \(f(x)=x^3-6x^2+9x+1\).
- \(f'(x)=3x^2-12x+9\).
- \(f''(x)=6x-12\).
- Solve \(f''(x)=0\): \(x=2\).
- For \(x<2\), \(f''(x)<0\); for \(x>2\), \(f''(x)>0\). Concavity changes.
- \(f(2)=3\).
The point of inflection is \((2,3)\).
Horizontal point of inflection
This occurs when \(f'(c)=0\) and \(f''\) changes sign at \(c\).
\[ f'(c)=0 \quad\text{and}\quad f''(x)\text{ changes sign at }c \]
Concavity and inflection
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Horizontal point of inflection
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