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Concave, Inflection point

A function is concave upward where its derivative is increasing. A function is concave downward where its derivative is decreasing.

concave upward
concave
              downward
Concave upoard
Concave downward


A point of inflection is a point where the function changes from being concave upward to being concave downward, or vice versa. A curve is concave upward where the parabola of best fit is concave upward and concave downward where the parabola of best fit is concave downward.

parabola of
                best fit
Point of inflection and parabola of best fit

The sign of the second derivative is a test for concavity. If f''(x) > 0, then f'  is increasing, so the slopes of the tangent lines increase as you go from left to right. Likewise, if  f''(x) < 0, then the slopes of the tangent lines decrease as you go from left to right. The two cases are shown here:

If  f''(x) > 0 then the graph rests on top of its tangent lines, while if f''(x) < 0 then the tangent lines rest on top of the graph. In the graph below, the tangent line at the inflection point  (1, -1) lies above the graph on one side and below the graph on the other.  


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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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