A

point of inflection of a graph is where the concavity of that graph changes. On

particle- or

object-

motion graphs, these points are where the

acceleration reverses.
A

point of inflection is a zero of the

double derivative (something that is not always easily-found). Take the

derivative of the derivative of the original function, set it equal to

zero, and solve for

X.

**Example:**
If the original equation is given as:

F(X) = X^4 - 4X^3 + 4

Then the double derivative is given as:

F``(X) = 12X^2 - 24X

Simplified, set equal to , and factored:12X^2 - 24X = 0
X^2 - 2X = 0 (divide both sides by 12)
X(X - 2) = 0 (factor)

So F``(X) = 0 when **X = 0 or 2**.

By plugging 0 and 2 into the original

F(X) equation, we find that the

points of inflection are

**(0, 4)** and

**(2, -12)**.
In other words, when X = 0, the

concavity flips around and curves the graph, making it

concave-

down, and when X = 2, the

concavity reverses once more, making it

concave-

up.

**Try it for yourself**: Plug

*X^4 - 4X^3 + 4* into your

graphing calculater and see how much better you can understand the

curves in a graph from the

points of inflection.