1. **State the problem:** We are given two functions: $$f(x) = 3x^4$$ $$g(x) = 2 \cdot 5^x$$ We need to determine which statements about these functions are true. 2. **Understand the functions:** - $f(x) = 3x^4$ is a polynomial function of degree 4, which grows quickly for large $|x|$. - $g(x) = 2 \cdot 5^x$ is an exponential function with base 5, which grows faster than any polynomial as $x \to \infty$. 3. **Compare values at specific points:** - At $x=0$: $$f(0) = 3 \cdot 0^4 = 0$$ $$g(0) = 2 \cdot 5^0 = 2 \cdot 1 = 2$$ So, $g(0) > f(0)$. - At $x=1$: $$f(1) = 3 \cdot 1^4 = 3$$ $$g(1) = 2 \cdot 5^1 = 10$$ So, $g(1) > f(1)$. - At $x=2$: $$f(2) = 3 \cdot 2^4 = 3 \cdot 16 = 48$$ $$g(2) = 2 \cdot 5^2 = 2 \cdot 25 = 50$$ So, $g(2) > f(2)$. - At $x=3$: $$f(3) = 3 \cdot 3^4 = 3 \cdot 81 = 243$$ $$g(3) = 2 \cdot 5^3 = 2 \cdot 125 = 250$$ So, $g(3) > f(3)$. 4. **Behavior for large $x$:** Since $g(x)$ is exponential and $f(x)$ is polynomial, for very large $x$, $g(x)$ will always be greater than $f(x)$. 5. **Behavior for negative $x$:** - For negative $x$, $f(x) = 3x^4$ is always positive because $x^4$ is positive. - For $g(x) = 2 \cdot 5^x$, since $5^x$ decreases as $x$ becomes more negative, $g(x)$ approaches 0 but remains positive. 6. **Summary:** - $g(x)$ is greater than $f(x)$ for $x \geq 0$. - Both functions are positive for all real $x$. **Final conclusion:** The statement "$g(x)$ is greater than $f(x)$ for all $x \geq 0$" is true.