University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Section 4.1 - Extreme Values of Functions - Exercises - Page 216: 82

Answer

If there are no endpoints and no critical points, the function will have no extrema (see example below).

Work Step by Step

If a continuous function has an extremum, it will be in an endpoint or a critical point. If there are no endpoints and no critical points, the function will have no extrema. An example is any linear function with nonzero slope over an open interval. For example, take $f(x)=x$, for $x\in(0,1)$. There is no point P in the interval that would be a minimum/maximum, as there are always points above (right of P) and below (left of P), however small an interval we choose around P. The derivative is $f'(x)=1$ for all x (never 0, defined everywhere on the interval). So, no critical points. Also, being defined on an an open interval, there are no endpoints.
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