TI-Nspire CX: Inflection Point Calculator Solver

What is an Inflection Point?

Inflections Points are Points of Largest Slope or Smallest Slope on a Graph.
You can also think of Inflections Points as Points where the graph changes from right turn to left turn or vice versa.

How do you find Inflection Points?

Inflection Point are found by finding sign changes of the 2. derivative. Yes, it does take Calculus.

What is an Inflection Point on a Graph?

At Inflections Points a graph changes its concavity. From concave up to concave or vice versa as shown in image below.

What is the definition of Concavity and Infection Points in Calculus?

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 some point x = c, then there is an Inflection Point located at x = c on the graph.

The above image shows an Inflection Point. It occurs when concavity changes. It is the Point of Steepest Slope.

Inflection Point Calculator

How do I find Inflection Points?

Inflections Points are found by finding the zeros of the 2. derivative AND ensuring a change a sign at those zeros.

Example (simple): If

f(x)=x3+5

then

f′(x)=3x2

and

f′′(x)=6x=0

when x=0. Since $f^{''} (x) = 6 x$ changes sign at x=0 then f(x) has an Inflection Point x=0. That was not complicated 😉

Example (No Inflection Point): If

f(x)=x2+5

then

f′(x)=2x

and

f′′(x)=2

Since $f^{''} (x) = 2$ never changes sign f(x) has no Inflection Point.
That was not complicated either 😉

Example (advanced): If

f(x)=x4−6x3

then

f′(x)=4x3−18x2

and

f′′(x)=12x2−36x=12x∗(x−3)

Since $f^{''} (x)$ changes sign at both $x = 0, 3$ f(x) has Inflection Points at $x = 0, 3$ .
Now try the above Infection Point Calculator a few times.

Why the 2. Derivative? It is all about the Rate of Change of the Change...

You may need a coffee before digesting this.. Here we go:
The 1. derivative f’ tells us whether f increase (f’ is positive) or decreases (f’ is negative).

Attention: There are 2 types of increases!

Type1-Increase:
Exponential-type growth occurs when f increases at an increasing rate. Think of an initial spread of a virus or initial spread of rumors, they spread fast. The increase is increasing which produces a concave up graph. Here, the 2. derivative or the rate of change of the increase is positive.

If you drove along the graph from left to right you would be turning left.

Type2-Increase:
Graph is leveling off. This occurs when f increases at an decreasing rate. Think of the final spread of a virus or the final spread of rumors. Saturation occurs. The increase is decreasing which causes a concave down graph. The 2. derivative or the rate of change of the increase is negative.

If you drove along the graph from left to right you would be turning right.

Now: The instant when f changes from increasing at an increasing rate to increasing at a decreasing rate constitutes a point of inflection. This point is (and must be) the point of greatest increase.

If you drove along the graph from left to right you would be changing from a left turn to a right turn at the point of inflection. At that point your steering wheel would be straight for a brief moment.

The same logic holds true for decreasing graphs: A point of inflection occurs when a graph decreasing at an increasing rate (concave down) changes to decreasing at a decreasing rate (concave up) [= the point of greatest decrease) or vice versa.

In conclusion: Points of Inflection occur when concavity changes this happens at the points of steepest in- or decrease.