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: 84

Answer

$(a) \quad$ f(x) can have 0,1, or 2 critical points. See examples below. $(b)\quad$ f(x) either has two or no extreme values.

Work Step by Step

$f(x)=ax^{3}+bx^{2}+cx+d$ $f'(x)=3ax^{2}+2bx+c$ is a polynomial, so no critical points are caused by $f'(x)$ not being defined. $f'(x)$ is a quadratic polynomial, and can have 0, 1, or 2 real zeros. This means that f can have 0,1, or 2 critical points. Examples (see graphs below): $f'(x)=x^{2}-1 \quad$ has two zeros. A function with this derivative could be $f(x)=\displaystyle \frac{x^{3}}{3}-x$ $f'(x)=(x-1)^{2}=x^{2}-2x+1 \quad\quad$ has one zero. A function with this derivative could be $f(x)=\displaystyle \frac{x^{3}}{3}-x^{2}+x$ Note that there are no extreme values in this case. $f'(x)=x^{2}+1 \quad$ has no zeros. A function with this derivative could be $f(x)=\displaystyle \frac{x^{3}}{3}+x$ Note that there are no extreme values in this case. $(b)$ From the above examples, we see that f(x) can have 2 or no extreme values. If it doesn't have two critical points, it has no extrema.
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