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: 49

Answer

The maximum value of the function is $7$ is at attained when $x=\pm\sqrt{2}$.

Work Step by Step

As suggested, we would use the substitution $t = x^2$ to get $f(t) = 3+4t-t^2$. To find the maximum of this quadratic function, we find the vertex using the formulas derived in the chapter, mainly $h=\frac{-b}{2a}$ and $k = f(h)$ to get $h=2$ and $k = f(2) = 7.$ We note that this would be the solution for the new equation $f(t)$ rather than $f(x)$. To get the solution for $f(x)$, we plug back in $t=x^2$ to get $x=\pm\sqrt{t}$ hence the maximum value of the function is $7$ and is at attained at $x=\pm\sqrt{2}$. Attached is the graph of both $f(x)$ and $f(t)$ to illustrate the results with the points above marked as well.
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