13 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W...

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x y y = f ( x ) x 0 ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 13 APPLICATIONS OF DIFFERENTIATION 13.1. Second Derivatives Recall that for a function y = f ( x ), the derivative f 0 ( x ) represents the slope of the tangent. It is easy to see from a picture that if the derivative f 0 ( x ) > 0, then the function f ( x ) is increasing; in other words, f ( x ) increases in value as x increases. On the other hand, if the derivative f 0 ( x ) < 0, then the function f ( x ) is decreasing; in other words, f ( x ) decreases in value as x increases. We are interested in the case when the derivative f 0 ( x ) = 0. Values x = x 0 such that f 0 ( x 0 ) = 0 are called stationary points. Let us introduce the second derivative f 0 ( x ) of the function f ( x ). This is deFned to be the derivative of the derivative f 0 ( x ). With the same reasoning as before but applied to the function f 0 ( x ) instead of the function f ( x ), we conclude that if the second derivative f 0 ( x ) > 0, then the derivative f 0 ( x )is increasing. Similarly, if the second derivative f 0 ( x ) < 0, then the derivative f 0 ( x ) is decreasing. Suppose that f 0 ( x 0 )=0and f 0 ( x 0 ) < 0. The condition f 0 ( x 0 ) < 0 tells us that the derivative f 0 ( x ) is decreasing near the point x = x 0 . Since f 0 ( x 0 ) = 0, this suggests that f 0 ( x ) > 0 when x is a little smaller than x 0 , and that f 0 ( x ) < 0 when x is a little greater than x 0 , as indicated in the picture below. This chapter was written at Macquarie University in 1999.

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x y y = f ( x ) x 0 x y x 1 x 2 13–2 W W L Chen and X T Duong : Elementary Mathematics In this case, we say that the function has a local maximum at the point x = x 0 . This means that if we restrict our attention to real values x near enough to the point x = x 0 , then f ( x ) f ( x 0 ) for all such real values x . LOCAL MAXIMUM. Suppose that f 0 ( x 0 )=0 and f 0 ( x 0 ) < 0 . Then the function f ( x ) has a local maximum at the point x = x 0 . Suppose next that f 0 ( x 0 ) = 0 and f 0 ( x 0 ) > 0. The condition f 0 ( x 0 ) > 0 tells us that the derivative f 0 ( x ) is increasing near the point x = x 0 . Since f 0 ( x 0 ) = 0, this suggests that f 0 ( x ) < 0 when x is a little smaller than x 0 , and that f 0 ( x ) > 0 when x is a little greater than x 0 , as indicated in the picture below. In this case, we say that the function has a local minimum at the point x = x 0 . This means that if we restrict our attention to real values x near enough to the point x = x 0 , then f ( x ) f ( x 0 ) for all such real values x . LOCAL MINIMUM. Suppose that f 0 ( x 0 and f 0 ( x 0 ) > 0 . Then the function f ( x ) has a local minimum at the point x = x 0 . Remark. These stationary points are called local maxima or local minima because such points may not maximize or minimize the functions in question. Consider the picture below, with a local maximum at x = x 1 and a local minumum at x = x 2 .
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13 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W...

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