1. The problem is to analyze the function $f(n) = 4n^3 - 2n^2 + n - 5$. 2. This is a cubic polynomial function where each term is a power of $n$ multiplied by a coefficient. 3. To understand the behavior of $f(n)$, we can evaluate it for specific values or analyze its growth. 4. For example, to find $f(1)$, substitute $n=1$: $$f(1) = 4(1)^3 - 2(1)^2 + 1 - 5 = 4 - 2 + 1 - 5 = -2$$ 5. To find $f(0)$, substitute $n=0$: $$f(0) = 4(0)^3 - 2(0)^2 + 0 - 5 = -5$$ 6. The function is continuous and differentiable everywhere since it is a polynomial. 7. The derivative $f'(n)$ is: $$f'(n) = 12n^2 - 4n + 1$$ 8. This derivative can be used to find critical points and analyze increasing/decreasing behavior. Final answer: The function is $f(n) = 4n^3 - 2n^2 + n - 5$ with derivative $f'(n) = 12n^2 - 4n + 1$ which helps analyze its behavior.