Inflection Point
In short: Inflection point marks the exact spot on a smooth, unbroken curve where its bend flips from cup to cap or from cap to cup, which is where f” changes sign.
AP Calculus ABCalculusAP Calculus AB Unit 5
What is Inflection Point?
An inflection point is the spot where a curve stops bending one way and starts bending the other, like the middle of the letter S. Driving along it, you would turn the steering wheel left and then right, with the wheel passing through straight at the inflection point.
Formally, \((c, f(c))\) qualifies when \(f\) is continuous at \(c\) and \(f^{\prime\prime}\) switches sign there. Having \(f^{\prime\prime}(c) = 0\) only makes \(c\) a candidate; the sign must truly flip.
How to spot it
- A question asks where the graph changes from concave up to concave down, or the reverse.
- A rate of change reaches its largest or smallest value, as when growth stops speeding up and starts slowing.
- A graph of a cubic or an S-shaped curve is shown and one point seems to twist the bend.
- You are asked to “justify” your answer, which means showing a sign change of the second derivative.
The method
- Compute \(f^{\prime\prime}(x)\).
- Solve \(f^{\prime\prime}(x) = 0\) and also look for inputs where it is undefined but \(f\) is defined.
- Check the sign of \(f^{\prime\prime}\) on both sides of each candidate with test values.
- A sign change confirms the candidate; find its height \(f(c)\) to state the full coordinates.
Inflection Point: solved example
Problem
Find the inflection points of \(f(x) = x^4 – 4x^3\).
Solution
\(f^{\prime}(x) = 4x^3 – 12x^2\) and \(f^{\prime\prime}(x) = 12x^2 – 24x = 12x(x – 2)\). The candidates are \(x = 0\) and \(x = 2\).
Signs of \(f^{\prime\prime}\): at \(x = -1\) it is \(36 > 0\); at \(x = 1\) it is \(-12 < 0\); at \(x = 3\) it is \(36 > 0\). The sign changes at both candidates.
Heights: \(f(0) = 0\) and \(f(2) = 16 – 32 = -16\). The bend flips at \((0, 0)\) and \((2, -16)\).
Try it
Question 1
Does the graph of \(f(x) = x^4\) flip its bend at \(x = 0\)?
Show the answer
\(f^{\prime\prime}(x) = 12x^2\) equals 0 at \(x = 0\) but is positive on both sides. There is no sign change, so the answer is no.
Question 2
Where does the bend of \(g(x) = x^3 – 6x^2 + 4\) change direction?
Show the answer
\(g^{\prime\prime}(x) = 6x – 12\) changes sign at \(x = 2\), and \(g(2) = 8 – 24 + 4 = -12\). The point is \((2, -12)\).
Inflection Point vs. Critical Point
A critical point comes from the first derivative and signals a possible peak, valley or flat spot. An inflection point comes from the second derivative and signals a change in bending. On \(y = x^3\) the origin is both, yet on \(y = x^2\) the origin is critical only.
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Whole dictionary: A–Z · AP Calculus AB · Calculus