1. The problem asks which statement guarantees the existence of a number $c$ in the interval $[-2, 3]$ such that $f(c) = 10$. 2. Statement A: $f$ is increasing on $[-2, 3]$ with $f(-2) = 0$ and $f(3) = 20$. - Since $f$ is increasing and $10$ lies between $0$ and $20$, by the Intermediate Value Property for increasing functions, there must be some $c$ in $[-2, 3]$ with $f(c) = 10$. 3. Statement B: $f$ is increasing on $[-2, 3]$ with $f(-2) = 15$ and $f(3) = 30$. - Here, $10$ is not between $15$ and $30$, so no guarantee that $f(c) = 10$ for some $c$ in $[-2, 3]$. 4. Statement C: $f$ is continuous on $[-2, 3]$ with $f(-2) = 0$ and $f(3) = 20$. - By the Intermediate Value Theorem, since $f$ is continuous and $10$ lies between $0$ and $20$, there exists $c$ in $[-2, 3]$ such that $f(c) = 10$. 5. Statement D: $f$ is continuous on $[-2, 3]$ with $f(-2) = 15$ and $f(3) = 30$. - Since $10$ is not between $15$ and $30$, the Intermediate Value Theorem does not guarantee $f(c) = 10$. 6. Conclusion: Statements A and C guarantee the existence of $c$ such that $f(c) = 10$. Final answer: Statements A and C guarantee $f(c) = 10$ for some $c$ in $[-2, 3]$.