College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.1 - Page 344: 62



Work Step by Step

See p.340: Strategy for Solving Problems lnvolving Maximizing or Minimizing Quadratic Functions 1. $\quad $Read the problem carefully and decide which quantity is to be maximized or minimized. 2. $\quad $Use the conditions of the problem to express the quantity as a function in one variable. 3. $\quad $Rewrite the function in the form $f(x)=ax^{2}+bx+c$. 4. $\quad $Calculate $-\displaystyle \frac{b}{2a}$. If $a>0, f$ has a minimum of $f$($-\displaystyle \frac{b}{2a}$) at $x=-\displaystyle \frac{b}{2a}$. If $a<0, f$ has a maximum of $f$($-\displaystyle \frac{b}{2a}$) at $x==-\displaystyle \frac{b}{2a}$ 5. $\quad $Answer the question posed in the problem. ---------------------------- 1. Product is to be maximized 2. Let x be one of the numbers. Then, the other number is $20-x$ Their product, $f(x)=x(20-x)$ is a quadratic function. 3.$\quad f(x)=20x-x^{2}$ $f(x)=-x^{2}+20x$ $4.\qquad a=-1, b=20, c=0$ $-\displaystyle \frac{b}{2a}=-\frac{20}{2(-1)}=10$ 5. The product is maximum when $x=10.$ The other number is $20-10=10.$ The maximum product is $f(10)=10(10)=100$
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