Which of the two functions below has the smallest minimum $y$-value?
\[
\begin{array}{c}
f(x)=(x-13)^{4}-2 \\
g(x)=3 x^{3}+2
\end{array}
\]
A. $f(x)$
B. There is not enough information to determine
C. The extreme minimum $y$-value for $f(x)$ and $g(x)$ is $-\infty$
D. $g(x)$

Step 1 ：Find the derivative of the functions $f(x)$ and $g(x)$ to determine where they reach their minimum or maximum values.

Step 2 ：The derivative of $f(x)$ is $4(x-13)^3$. Setting this equal to zero gives $x=13$.

Step 3 ：Substitute $x=13$ back into $f(x)$ to find the corresponding $y$-value. This gives $y=-2$.

Step 4 ：The derivative of $g(x)$ is $9x^2$. Setting this equal to zero gives $x=0$.

Step 5 ：Substitute $x=0$ back into $g(x)$ to find the corresponding $y$-value. This gives $y=2$.

Step 6 ：Compare the minimum $y$-values of $f(x)$ and $g(x)$. The function $f(x)$ has the smallest minimum $y$-value.

Step 7 ：Final Answer: \(\boxed{\text{A. } f(x)}\)