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: \(\boxed{\text{A. } f(x)}\)