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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.
Mathematical terms can be useful to describe interpersonal situations as well.

When you're trying to get to know someone of the opposite sex, it can extend through a series of conversations where you just talk about totally random stuff. During this sequence, each person involved will generally give some implied signs of interest in what the other is talking about.

If the friendship is to go anywhere into the romantic territory, these signs generally don't stop for a while. If it's not going anywhere, though, and one side decides it's time to subtly put an end to things, the last such conversation will have a point where it is suddenly clear that the other person doesn't give a rat's ass what you're talking about, and is only perpetuating the conversation for the sake of politeness.

Maybe the shape of the conversation doesn't immediately change that much, just like the slope of the graphed equation doesn't change much when the point of inflection is passed. But things are about to take a turn for the worse, much like that graph turning over.

A point of inflection is, as Ryouga describes, a point on a graph where concavity changes (making it the second derivative equivalent of a local extremum). He's also correct that most points of inflection are zeroes of the second derivative. However, not every zero of the second derivative is an inflection point.

Take, for example, the function f(x)=x4. The first derivative is f'(x)=4x3, the second derivative is f''(x)=12x2. Finding the zeroes of the second derivative, 12x2=0, gives x=0. However, x=0 is not the location of an inflection point.

The second derivative, f''(x), is 12x2. The graph is concave up where the second derivative is positive - 12x2 is positive everywhere except at zero, so the graph is concave up on both sides of x=0. Thus, the concavity does not change, so it's a zero of the second derivative but it's not a point of inflection.

There are also inflection points where the second derivative is not zero. Take, for example, f(x)=x1/3. The first derivative is f'(x)=(1/3)x-2/3, the second derivative is f''(x)=(-2/9)x-5/3. The second derivative has no zeroes. However, for negative values of x, the graph is concave up, and for positive values of x, the graph is concave down. x=0 is the location of an inflection point (f''(0) is undefined). So, inflection points can not only occur where the second derivative is zero - they can also occur where the second derivative is undefined.

tdent informs me that you can tell whether a point which has f''(x)=0 is an infleciton point or not by looking at subsequent derivatives (third derivative, fourth derivative, etc.). If the next nonzero derivative (evaluated for that particular value of x) is an odd derivative (third, fifth, etc.), then the point is an inflection point - if the next nonzero derivative at that x is an even derivative (fourth, sixth, etc.), then it is not.

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