Question

question 17 of 25
which of the two functions below has the smallest minimum y-value?
$f(x) = x^5 - 2$
$g(x) = 3x^2 + 1$
a. $f(x)$
b. $g(x)$
c. there is not enough information to determine.
d. the $y$-values for $f(x)$and $g(x)$ both go to $-infty$.

Explanation:

Step1: Analyze \(f(x) = x^5 - 2\)

As \(x \to -\infty\), \(x^5 \to -\infty\), so \(f(x) \to -\infty\) (no minimum).

Step2: Analyze \(g(x) = 3x^2 + 1\)

\(x^2 \geq 0\), so \(3x^2 \geq 0\), thus \(g(x) \geq 1\) (minimum is 1).

Step3: Compare minimums

\(f(x)\) has no lower bound, so its "minimum" is smaller than \(g(x)\)'s minimum 1.

Answer:

A. \(f(x)\)