Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - 3.7 Optimization Problems - 3.7 Exercises - Page 264: 2


$(a,b) = (50, -50)$

Work Step by Step

Find two numbers whose difference is 100 and whose product is a minimum. Write out the equations $a-b=100$ and $y=ab$ Substitute $a=100+b$ $y=(100+b)b$ $y=100b + b^2$ The problem asks to optimize y, so differentiate and set the derivative $=0$ $y'=100+2b$ $-100=2b$ $b=-50$ Use the second derivative test to confirm that $b=-50$ is in fact a minimum. $y''= 2$ Concave up on a critical point confirms that $b=-50$ is a minimum. Therefore, $a=50$. $(a,b) = (50, -50)$
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