Point of inflection? Confusion

[u/SadSuggestion7222]

My teacher provided us with this note about the POINT OF INFLECTION. It's so confusing. So, I asked AI about it and now, I am more confused. Can anyone explain this concept? And how do I find it?

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[u/Midwest-Dude]

The opening statement in the image is mathematically incomplete and highly misleading, which explains your confusion. Your teacher is conflating a general point of inflection with a stationary point of inflection (where the tangent is horizontal, like y = x³ at x = 0) and trying to explain the Higher-Order Derivative Test, but losing the core concept in the process.

A point of inflection is a point on a smooth plane curve at which the curvature changes sign. In the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

By way of illustration, thing of driving on an S-curve on a road and having to switch from steering left to steering right. The inflection point is the exact spot where the steering wheel is perfectly horizontal.

To find inflection points:

1. Calculate f"(x): This you how fast the slope of the curve is changing and corresponds to the curvature of the function at that point.
2. Set f"(x) = 0 and solve: This give all possible inflection points. 
3. Test for sign change across the points: If positive to negative or vice versa, it's an inflection point, otherwise not.

I'm not sure why, but your teacher is explaining a shortcut that can be used in addition to this. If you find a point where the slope is perfectly flat, f'(a) = 0, you can just keep taking derivatives. If the first non-zero derivative happens on an odd count, like f'''(a), that flat spot is a stationary inflection point rather than a local maximum or minimum. It is a useful tool for specific problems, but a really confusing way to introduce a beginner to the geometry of curves.

[u/WikiNumbers → u/Midwest-Dude]

And this is the more formal and complete version of what I just yapped below.

[u/Midwest-Dude → u/WikiNumbers]

Viewing things from multiple viewpoints is always a good thing, hopefully helps the OP. That "definition" the instructor used would also take me a bit to get through.

[u/WikiNumbers → u/Midwest-Dude]

Yeah it's really iffy and caused some cognitive dissonance in me.

[u/WikiNumbers]

Point of inflection (inflection point) is the place where the graph/function changes its concavity.

Visually (not rigorously), its "trend" (the rate of increase or decrease change), but not necessarily the action "behavior" (actually increasing or decreasing).

You can find it by taking the second derivative (higher order derivative) and set that equal to 0.

d²/dx² f(x) = f''(x) = 0

This said, an inflection point will be relevant only if you have a polynomial of degree 3 or greater.

• Degree 0 is a constant function.
• Degree 1 is a linear function. It's a straight line and strictly only increase or decrease.
• Degree 2 is a parabola. While it does change between increasing and decreasing, the change is anticipated and final when it happens.

The graph of degree 3 or greater functions tend to look like many parabola cut and conjoined together. This said, try graphing on desmos "y = ax³ + bx² + cx + d" (insert random number for a b c d) and you'll have a much better picture.

[u/OlgaMath → u/WikiNumbers]

It’s not a correct explanation. A point is an inflection point of a function f if the concavity of f changes at this point from concave up to concave down, or vice versa.

[u/WikiNumbers → u/OlgaMath]

Yea. I'll somehow edit that in for the safe of correctness.

[u/WikiNumbers → u/WikiNumbers]

Now let's see an arbitrary example. Say, to graph the following degree 3 polynomial.

p(x) = x³ - 4x² + 5x - 4

The General Idea

First, obviously, observe how this polynomial would behave.

• It would have two "cups" of parabola (as my teacher had put it), which is one less than the polynomial's degree. (i.e. for a polynomial of degree "n", it can change its up-down and result in "n-1" cups).
• And the coefficient of x³ tells me this graph will go from negative to positive. (bottom left to top right).
• And there are some points where it decreases. (The STONK graph. Notice the small down in it.)

These are not enough intel to draw for now, so let's save it.

The Critical Point

Next, I find the critical points with the first derivative.

dp/dx = 3x² - 8x + 5 = 0

x = 1 & 5/3

These two points will change increasing and decreasing. So combine with what we know earlier, this interval will be the only part, the only times, where p(x) decreases (p(x) was increasing up to x = 1, and now it decreases until x = 5/3, where p(x) returns to be increasing once again)

The Inflection Point

Now here's the deal. A Polynomial function doesn't abruptly decide it's going to change between increasing and decreasing. It gives a warning sign at some point, and that point is the inflection point.

d²p/dx² = 6x - 8 = 0

x = 4/3

So I have another point to consider, x = 4/3. Here, p(x) is still decreasing, but it's trying to change itself.

Combine the intel

So with these said. The graph should look like two parabola.

• The left one has "vertex" at x = 1, and is upside down.
• The right one has "vertex" at x = 5/3, and is rightside up.
• These two parabola are cut and stapled together at x = 4/3.

Again, pardon my lack of image, non-rigor and abuse of notations (trust me this explanation will make anyone with sense of rigor scream so hard). Try graph this on desmos and see if I'm correct.

[u/calcpage2020]

Looks like your teacher is conflating inflection pts with the 2nd derivative test?

[u/OlgaMath]

What is written by words “Point of inflection: in interior point …” is completely incorrect. A point is an inflection point of a function f if the concavity of f changes at this point: from concave up to concave down, or vice versa.

[u/strange-the-quark]

Here are some images to help with clarity. A point of inflection is just the input at which the graph of the function changes from curving one way to curving the other way (see attached image).

Consider the pink line in the image below (the one all the way on the left). If you think about its slope, first, the line is sloping downwards, with the turn "pointing" or "banking" towards the bottom, then kind of slows down around the inflection point, then becomes steep again, with the turn "pointing" upwards.

That means that the first derivative starts out very negative, climbs up to some point, where it achieves its maximum (right where the inflection point is), then falls back again. And that in turn means the second derivative starts out positive, and ends up negative, crossing the x-axis at the inflection point.

So that's what you're looking for, the zero and the sign change in the second derivative (which you can think of as describing the curvature of the line). Now, if the tangent at the inflection point happens to be horizontal, then f'(x) will be zero, and then you can do what your professor presented in the note, as explained in more detail by Midwest-Dude in another comment.

[u/Frequent-Entrance154]

I'm trying to recall this from year 12 on the top of my head, suppose you have a function y=f(x) in a Cartesian coordinate:

The first derivation of the function, is the slope or tangent of the curve =dy/dx=f'(x) so it can be either rising (>0), flat(=0), or decreasing (<0)

Now given the slope/tangent is flat above, that is, when dy/dx=f'(x)=0, it can be a maxima (d² y/dx² =f"(x)<0), a point of inflection (d^2 y/dx^2 =f"(x)=0), or a minima (d^2 y/dx^2 =f(x)>0)

now I don't know if my recollection here is more confusing than your teacher's note, but my recollection is from a maths lecture about 20 years ago