Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.2 Exercises - Page 115: 79


No horizontal tangential line exists for this function.

Work Step by Step

Horizontal lines have, by definition, a slope of $m = 0$. Therefore, for this exercise, we are trying to find a value of $x$ where $f'(x) = 0$. Combining the Addition and Power rules, we can find $f'(x)$ as follows: $$f(x) = 3x + sinx + 2$$ $$f'(x) = 3x^{1-0} + (cosx) + 0$$ $$f'(x) = 3 + cosx$$ where the domain of $x$ is $[0, 2\pi]$. Solving for $f'(x) = 0$: $$0 = 3 + cosx$$ $$-3 = cosx$$ Since there is no solution to this equation within the domain of $x$ $[0, 2\pi]$, it can be concluded that no horizontal tangential line exists for this function.
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