1. **State the problem:** We have the function $f(x) = 4x^3 + 3x^2 + 13$ and want to find the values of $k$ such that there exists a $c \in [0,1]$ with $f(c) = k$ according to the Intermediate Value Theorem (IVT). 2. **Recall the Intermediate Value Theorem:** If $f$ is continuous on $[a,b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists $c \in [a,b]$ such that $f(c) = N$. 3. **Check continuity:** $f(x)$ is a polynomial, so it is continuous everywhere, including $[0,1]$. 4. **Evaluate $f$ at the endpoints:** $$f(0) = 4(0)^3 + 3(0)^2 + 13 = 13$$ $$f(1) = 4(1)^3 + 3(1)^2 + 13 = 4 + 3 + 13 = 20$$ 5. **Apply IVT:** For any $k$ between $f(0)$ and $f(1)$, i.e., between 13 and 20, there exists $c \in [0,1]$ such that $f(c) = k$. 6. **Final answer:** The values of $k$ are those satisfying $$13 \leq k \leq 20$$