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: 58

Answer

The graph has no horizontal tangents.

Work Step by Step

To find the slope of the tangents, we find the derivative first: $y'=(x^3+x)'=(x^3)'+(x)'=3x^2+1$ Horizontal means a slope of $0$ which indicates that the derivative should equal $0$: $y'=0\rightarrow3x^2+1=0\rightarrow x^2=-\frac{1}{3}$ Since over the domain of real numbers,$x^2$ cannot be negative then the above equation has no solution hence the derivative is never equal to $0$ and therefore there are no horizontal tangents.
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