Given the functions $f(x) = 5x^4$ and $g(x) = 6 \cdot 3^x$, let's analyze the given options:

Understanding the Functions:

1. $f(x) = 5x^4$:

   • This is a polynomial function of degree 4. As $x$ increases, $f(x)$ grows rapidly, particularly for large positive or negative values of $x$.

2. $g(x) = 6 \cdot 3^x$:

   • This is an exponential function with base 3. Exponential functions grow much faster than polynomial functions as $x$ increases.

Comparing Growth Rates:

• Exponential functions like $g(x) = 6 \cdot 3^x$ tend to grow faster than polynomial functions like $f(x) = 5x^4$ as $x$ becomes large.

Possible True Statements:

1. $f(x) > g(x)$ for all $x$.
2. $f(x) < g(x)$ for all $x$.
3. $f(x) = g(x)$ for some $x$.
4. $f(x) = g(x)$ for all $x$.
5. $f(x) > g(x)$ for large $x$.
6. $g(x) > f(x)$ for large $x$.

Evaluation:

• For large values of $x$, $g(x) = 6 \cdot 3^x$ will eventually outgrow $f(x) = 5x^4$, because exponential growth outpaces polynomial growth.

Therefore, the true statement is:

• $g(x) > f(x)$ for large $x$.

Would you like more details on this comparison or have any questions? Here are five related questions:

1. How do polynomial and exponential functions compare in growth rate?
2. What is the general form of an exponential function?
3. Can a polynomial ever outgrow an exponential function for any value of $x$?
4. What happens to $f(x)$ and $g(x)$ as $x$ approaches infinity?
5. How can you determine when $f(x) = g(x)$ for specific values of $x$?

Tip: When comparing different types of functions, always consider their long-term behavior, particularly as $x$ approaches infinity.