Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 4 - Applications of the Derivative - 4.4 The Shape of a Graph - Exercises - Page 194: 5


The inflection points are at $x=0, \pm\sqrt 3$. Concave up on $x\lt -\sqrt 3$ and $0\lt x\lt \sqrt 3$. Concave down on $x\gt \sqrt 3$.

Work Step by Step

We have $$y=10x^3-x^5, \quad y'=30x^2-5x^4, \quad y''=60x-20x^3$$ The inflection points occur when $y''=60x-20x^3=20x(3-x^2)=0$, that is $x=0, \pm\sqrt 3$. Concave up when $y''\gt 0$, which occurs at $x\lt -\sqrt 3$ and $0\lt x\lt \sqrt 3$. Concave down when $y''\lt 0$, which occurs at $x\gt \sqrt 3$.
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