An
*inflection point*
is a point
x_{0}
on the curve where concavity changes from concave up to concave down or vice versa.

Let's illustrate the above with an example. Consider the function shown in the figure.

From figure it follows that on the interval
(−∞,
x_{0})
the graph of the function is convex up (or concave down). On the interval
(x_{0},
∞)
- convex down (or concave up). The point
(x_{0},
y_{0})
is called an inflection point.

The intervals of convexity (concavity) of a function can easily be found by using the following theorem:

If the second derivative of the function is positive on certain interval, then the graph of the function is concave up on this interval. If it's negative - concave down. I.e.:

It also should be noted, than at inflection points the second derivative of the function is zero and changed its sign when passing through such points.

Our online calculator based on Woflram Alpha system allows you to find inflection points of the function with step by step solution.

Inflection points calculator

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