1. **State the problem:** We are given a function $f$ defined on intervals $[0,8]$ and $[2,8]$ with properties that satisfy Rolle's Theorem and the Mean Value Theorem (MVT). We need to estimate values of $c$ where the derivative $f'(c)$ meets certain conditions and interpret the theorems' conclusions. 2. **Rolle's Theorem (part a):** Rolle's Theorem states that if a function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c$ in $(a,b)$ such that $$f'(c) = 0.$$ 3. **Apply Rolle's Theorem:** Given $f(0) = f(8) = 0$ and $f$ is continuous and differentiable on $[0,8]$, there must be some $c$ in $(0,8)$ with $f'(c) = 0$. The graph shows a vertex at approximately $x=4$, where the tangent is horizontal, so $$c = 4.$$ 4. **Interpretation of Rolle's Theorem (part b):** The conclusion means there is a point $c$ where the tangent line is horizontal (slope zero). So the correct interpretation is: "There is a value of $c$ in $(0,8)$ such that the tangent line to $f$ at $x=c$ is horizontal." 5. **Mean Value Theorem (part c):** MVT states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c$ in $(a,b)$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$ 6. **Apply MVT:** On $[2,8]$, $f$ is continuous and differentiable. The average rate of change is $$\frac{f(8) - f(2)}{8 - 2}.$$ From the graph, the slope of the secant line between $(2,f(2))$ and $(8,f(8))$ is positive and the tangent line with this slope occurs near $c=5$ (not $11$ as stated). Since $11$ is outside $(2,8)$, the correct estimate should be near $c=5$. 7. **Interpretation of MVT (part d):** The conclusion means there is a $c$ where the tangent line is parallel to the secant line through $(2,f(2))$ and $(8,f(8))$. The best interpretation is: "There is a value of $c$ in $(2,8)$ such that the tangent line to $f$ at $x=c$ is parallel to the secant line passing through $(2,f(2))$ and $(8,f(8))$." **Final answers:** - (a) $c = 4$ - (b) Tangent line at $c$ is horizontal - (c) $c \approx 5$ (corrected estimate) - (d) Tangent line at $c$ is parallel to secant line