College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.1 - Quadratic Functions and Models - 3.1 Exercises - Page 288: 50


The minimum value is $-14$ attained when $x=-\sqrt[3] 2$.

Work Step by Step

As suggested, we use the substitution $t=x^3$ to get $f(x) = 2 + 16t+4t^2$ which is a quadratic function of the form $f(t) = at^2+bt+c$. To find the minimum value, we find the vertex using the methods outlined in the chapter; mainly, $h=-\frac{b}{2a}$ and $k=f(h)$ with the vertex being the point $(h, k).$ We plug the values to get $h=-2$ and $k=-14.$ We note that this would represent the set of values of $f(t)$ rather than the required $f(x)$. To get the values for $f(x)$, we plug in $t=x^3$ to get that $x=\sqrt[3]{t}$ and hence the minimum value of $f(x)$ is $-14$ attained when $x=-\sqrt[3]{2}$. Attached below is a graph illustrating the curves mentioned above and the respective minimum points.
Small 1560869317
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.