1. The problem asks which condition guarantees that the tangent line approximation at $x=5.25$ is an overestimate of $f(5.25)$. 2. Recall that the tangent line approximation at $x=a$ is given by: $$L(x) = f(a) + f'(a)(x - a)$$ This linear approximation is used to estimate $f(x)$ near $x=a$. 3. The accuracy of this approximation depends on the concavity of $f$ on the interval. If the graph of $f$ is concave down on the interval, the function lies below its tangent line, so the tangent line overestimates the function. 4. Mathematically, concavity is determined by the second derivative $f''(x)$: - If $f''(x) < 0$ on the interval, $f$ is concave down. - If $f''(x) > 0$ on the interval, $f$ is concave up. 5. Since the problem asks for the condition guaranteeing the tangent line approximation is an overestimate at $x=5.25$, the correct condition is that $f$ is concave down on $[5, 5.25]$. 6. Therefore, the correct answer is option C: The graph of the function $f$ is concave down on the interval $5 \leq x \leq 5.25$. Final answer: C