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The first derivative can provide very useful information about the behavior of a graph. This information can be used to draw rough sketches of what a function might look like. The second derivative, f''(x), can provide even more information about the function to help refine the sketches even further.
Consider the following graph of f on the closed interval [a, c]:
It is clear that f (x) is increasing on [a, c]. However, its behavior prior to point b seems to be somehow different from its behavior after point b.
A section of the graph of f (x) is considered to be concave up if its slope increases as x increases. This is the same as saying that the derivative increases as x increases. A section of the graph of f (x) is considered to be concave down if its slope decreases as x increases. This is the same as saying that the derivative decreases as x increases.
In the graph above, the segment on the interval (a, b) is concave up, while the segment on the interval (b, c) is concave down This can be seen be observing the tangent lines below:
The point b is known as a point of inflection because the concavity of the graph changes there. Any point where the graph goes from concave up to concave down, or concave down to concave up, is an inflection point.
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