Key takeaway: An inflection point is a point on a continuous curve where the concavity changes from up to down or from down to up, which happens where the second derivative changes sign.
Grade 12 – AP Calculus ABAP Calculus AB Unit 5Calculus
Definition of Inflection Point
An inflection point is where a curve switches the way it bends. Before the point it may open like a cup and after it like a cap, or the other way around. The tangent line there crosses through the curve instead of staying on one side.
At such a spot the slope stops increasing and starts decreasing, or the reverse, so the slope itself reaches a local maximum or minimum. Candidates come from the second derivative, but each one must pass a sign check.
Formula and notation
The point \((k,\,v(k))\) is where the bend flips when \(v\) is continuous at \(k\) and \(v^{\prime\prime}\) changes sign at \(k\). Candidates satisfy
\[ v^{\prime\prime}(k)=0\quad\text{or}\quad v^{\prime\prime}(k)\text{ undefined}. \]
A sign change on either side confirms the candidate; no change rejects it.
Key facts
- The second derivative must change sign; reaching zero alone is not sufficient.
- At such a point the tangent line crosses the curve.
- The first derivative has a local extreme value there.
- A function can have several of them, or none, as with a parabola.
Inflection Point example
Find the inflection points of \(f(x)=x^5-10x^3\).
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Differentiate twice: \(f^{\prime}(x)=5x^4-30x^2\) and \(f^{\prime\prime}(x)=20x^3-60x=20x(x^2-3)\).
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The second derivative vanishes at \(x=0\), \(x=\sqrt{3}\) and \(x=-\sqrt{3}\).
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Sample signs: \(f^{\prime\prime}(-2)=-40\), \(f^{\prime\prime}(-1)=40\), \(f^{\prime\prime}(1)=-40\), \(f^{\prime\prime}(2)=40\). The sign flips at all three zeros.
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Heights: \(f(0)=0\) and \(f(\pm\sqrt{3})=\pm(9\sqrt{3}-30\sqrt{3})=\mp 21\sqrt{3}\).
The curve has three of them: \((0,0)\), \((\sqrt{3},-21\sqrt{3})\approx(1.73,-36.4)\) and \((-\sqrt{3},21\sqrt{3})\approx(-1.73,36.4)\).
Common mistake
Watch out
A student finds \(u^{\prime\prime}(0)=0\) for \(u(x)=x^6\) and announces an inflection point at the origin.
Why it is wrong
Here \(u^{\prime\prime}(x)=30x^4\) stays nonnegative around 0, so the concavity never changes. The origin is a minimum, not a switch of bend. Zero is only a candidate; test the sign on each side before concluding.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus