1. **State the problem:** We know for a function $p(x)$, that the first derivative at $x=5$ is zero, i.e., $p'(5) = 0$, and the second derivative at $x=5$ is negative, i.e., $p''(5) = -3$. 2. **Interpret these values:** The derivative $p'(5) = 0$ suggests that $x=5$ is a critical point, potentially a maximum, minimum, or inflection point. 3. **Use the second derivative test:** If $p''(5) > 0$, then $p(5)$ is a local minimum; if $p''(5) < 0$, then $p(5)$ is a local maximum; if $p''(5) = 0$, the test is inconclusive. 4. **Evaluate $p''(5)$:** Since $p''(5) = -3 < 0$, the second derivative test indicates a local maximum at $x=5$. **Final answer:** The function has a local maximum at $x=5$ corresponding to option D.