University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 3 - Section 3.3 - Differentiation Rules - Exercises - Page 136: 60


$a=-3, b=2$ and $c=1$.

Work Step by Step

$y=f(x)=x^2+ax+b$ and $y=g(x)=cx-x^2$ 1) Since $g(x)$ has a tangent line at $(1,0)$, the curve $g(x)$ passes through the point $(1,0)$: $$c\times1-1^2=0$$ $$c-1=0$$ $$c=1$$ Therefore, the equation of $g(x)$ is $g(x)=x-x^2$ 2) Find $f'(x)$ and $g'(x)$: $$f'(x)=2x+a$$ $$g'(x)=1-2x$$ - At $(1,0)$, the slope of the tangent line to $f(x)$ and $g(x)$ is $$f'(1)=2\times1+a=2+a$$ $$g'(1)=1-2\times1=1-2=-1$$ Since $f(x)$ and $g(x)$ have the same tangent line at $(1,0)$, these two slopes must be equal. $$2+a=-1$$ $$a=-3$$ 3) Finally, since $f(x)$ has a tangent line at $(1,0)$, $f(x)$ must pass through the point $(1,0)$: $$1^2+a\times1+b=0$$ $$1+a+b=0$$ Substitute $a=-3$ here: $$1-3+b=0$$ $$-2+b=0$$ $$b=2$$
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