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Understanding Concavity

A differentiable function on some interval is said to be concave up if is increasing and concave down if is decreasing. If is constant, then the function has no concavity. Points where a function changes concavity are called inflection points.
The red line is the tangent to the curve at and the dashed blue line is the tangent to the curve a little to the right of .

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To visualize the idea of concavity using the first derivative, consider the tangent line at a point. Recall that the slope of the tangent line is precisely the derivative. As you move along an interval, if the slope of the line is increasing, then is increasing and so the function is concave up. Similarly, if the slope of the line is decreasing, then is decreasing and so the function is concave down.
In this Demonstration, the tangent line at is drawn in red. The tangent line at is denoted by a dashed blue line. If never changes sign twice in an interval .15 units wide or smaller, as is the case in all the examples considered by this Demonstration, then whenever the blue line has a larger slope than the red line, the derivative is increasing from to and the function is concave up on that interval. Likewise, whenever the blue line has a smaller slope than the red line, the derivative is decreasing from to and the function is concave down on that interval.
In practice, we use the second derivative test to check concavity. The second derivative test says that a function is concave up when and concave down when This follows directly from the definition as the is concave up when is increasing and is increasing when its derivative is positive. Similarly is concave down when is decreasing, which occurs when .
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