CALCULUS Understanding Its Concepts and Methods

Concave, Inflection point

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

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.

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.