AP Calculus Review: Inflection Points

What are inflection points, and how do you find them ? This article explains what you need to know about inflection points for the AP Calculus exams .

Inflection Points and Concavity

An inflection point is a point in a graph at which the concave shape changes .
Graph showing concave down, inflection points, and concave up.
so lashkar-e-taiba ’ s talk a little about concavity first .


concavity is about curving.

We say that a function yttrium = f ( ten ) is concave up (CU) on a given interval if the graph of the function always lies above its tangent lines on that interval. In other words, if you draw a tangent line at any given point, then the graph seems to curve upwards, away from the line .
parabola, concave up
conversely, a function is concave down (CD) on a given time interval if the graph of the function always lies below its tangent lines on that time interval. That is the graph seems to curve downwards, away from its tangent line at any given point .
parabola, concave down
This say may help you to remember which matchless is which : Concave up looks like a cup ; concave down looks like a frown .


Some functions, like the parabolas shown above, only have one kind of concave shape. But more interesting functions may have a mixture of the two. The graph might be concave down in one interval and concave up in the adjacent. Some functions even switch back and forth between concave up and concave down infinitely many times !
Any period at which concavity changes ( from CU to CD or from candle to CU ) is call an inflection point for the function .
For example, a parabola fluorine ( x ) = ax2 + bx + hundred has no modulation points, because its graph is always concave up or concave down .

Inflection Points and Derivatives

inflection points may be unmanageable to spot on the graph itself. So we must rely on calculus to find them .
basically, it boils depressed to the second base derivative .

  • If f ” > 0 on an interval, then f is concave up on that interval.
  • If f ” < 0 on an interval, then f is concave down on that interval.
  • If f ” changes sign (from positive to negative, or from negative to positive) at a point x = c, then there is an inflection point located at x = c on the graph. In particular, the point (c, f(c)) is an inflection point for the function f.

here ’ s a beneficial rule of flick. Look for potential prosody points by setting farad ” = 0. That ’ south because the alone way for f ” to change signs is to pass through the “ neutral territory ” of 0 ( well, assuming the change is continuous, anyhow ) .

Why the Second Derivative?

so what ’ s sol particular about the second derived function that makes it the go-to creature for concavity and prosody ?
It all has to do with change — in this event, the changing slope of a graph as it curves up or down .
In ordering for the graph to be concave up, it must always increase slope as it progresses from left to right. now remember, the inaugural derivative measures slope. indeed farad ‘ must be increasing whenever the graph is concave up. The final examination piece of the puzzle is the observation that increasing functions have positive derivatives. consequently, the derivative instrument of degree fahrenheit ‘, that is, f ”, must be cocksure !
analogously, if the graph is concave down, then degree fahrenheit ‘ must be a decreasing function. Hence, concave down means that fluorine ” must be veto.

Finding Inflection Points

Let ’ s see by exercise how to locate the inflection points of a graph. Along the way, we ’ ll find out where the serve is concave up and concave down .
Let polynomial, sixth degree

. Find all points of inflection and intervals of concavity for f.


. Find all points of inflection and intervals of concave shape for farad. Our first task is to find the first and second derivatives .
first and second derivative of an example function
following, set the second gear derived function equal to zero and solve. Because farad ” is a polynomial, we have to factor it to find its roots .
polynomial factoring example
There are three solutions, x = 0, 4, and -3. These are not necessarily wholly going to be prosody points, though ! We have to find out how the concave shape changes from interval to interval foremost .
Let ’ s set up table to figure out what happens in each interval. Within each interval, choose a sample point to plug into fluorine ” to determine the concave shape .

Interval Sample f ” Concavity
(-∞, -3) -4 128 (> 0) CU
(-3, 0) -1 -10 (< 0) CD
(0, 4) 1 -12 (< 0) CD
(4, ∞) 5 200 (> 0) CU

therefore, the function is concave up on ( -∞, -3 ) U ( 4, ∞ ). It ’ s concave down on ( -3, 0 ) U ( 0, 4 ). Notice that there is no change in concave shape at x = 0 .
That means that the entirely prosody points are at x = -3 and 4. Plug each of those points into the original function farad ( ten ) to find their corresponding y-coordinates. ( A calculator can help out greatly here ! )

  • f(-3) = -33.55
  • f(4) = -138.67

thus the two inflection points for this function are : ( -3, -33.55 ) and ( 4, -138.67 ). Let ’ s take a expect at the graph and see if we can spot how the concave shape changes at those points.

Graph of function showing concavity and inflection.


  • An inflection point is a point on the graph of a function at which the concavity changes.
  • Points of inflection can occur where the second derivative is zero. In other words, solve f ” = 0 to find the potential inflection points.
  • Even if f ”(c) = 0, you can’t conclude that there is an inflection at x = c. First you have to determine whether the concavity actually changes at that point.

For more information about concave shape, inflections, and other relate topics, check out AP Calculus Review : Functions and their graph


  • Shaun Ault
    Shaun Ault
    Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 ( Go Bucks ! ! ). He received his BA in Mathematics with a minor in calculator science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music — about equally a lot as mathematics ! — and he ( thinks he ) can play piano, guitar, and bass. Shaun has taught and tutor students in mathematics for about a ten, and hopes his feel can help you to succeed !
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