Question

question 9 of 10
which of the two functions below has the largest maximum y-value?
$f(x) = -x^4 - 2$
$g(x) = -3x^3 + 2$

a. $g(x)$
b. there is not enough information to determine
c. $f(x)$
d. the extreme maximum y-value for both $f(x)$ and $g(x)$ is $\infty$

Explanation:

Step1: Analyze \(f(x) = -x^4 - 2\)

Since \(x^4 \geq 0\) for all real \(x\), \(-x^4 \leq 0\). Thus \(f(x) \leq 0 - 2 = -2\). Max \(y\)-value of \(f(x)\) is \(-2\).

Step2: Analyze \(g(x) = -3x^3 + 2\)

As \(x \to -\infty\), \(x^3 \to -\infty\), so \(-3x^3 \to +\infty\). Thus \(g(x) \to +\infty\), meaning it has no finite maximum.

Step3: Compare maxima

\(g(x)\) has no upper bound (grows to infinity), while \(f(x)\) has a maximum of \(-2\). So \(g(x)\) has a larger maximum.

Answer:

A. \(g(x)\)