1. **Problem Statement:** We are given that the first derivative $k'(c) = 0$ and the second derivative $k''(c) = 0$ at the point $x = c$. We want to determine what can be concluded about the function $k(x)$ at this point. 2. **Recall the derivative tests:** - The first derivative test uses $k'(c)$ to find critical points where the function could have local maxima, minima, or neither. - The second derivative test uses $k''(c)$ to determine the concavity at $x = c$ and helps classify the critical point: - If $k''(c) > 0$, $k(c)$ is a local minimum. - If $k''(c) < 0$, $k(c)$ is a local maximum. - If $k''(c) = 0$, the test is inconclusive. 3. **Given:** $k'(c) = 0$ means $x = c$ is a critical point. 4. **Given:** $k''(c) = 0$ means the second derivative test cannot classify the critical point. 5. **Conclusion:** Since both derivatives are zero, the second derivative test is inconclusive. The function could have a local max, local min, inflection point, or none of these at $x = c$. More information or higher-order derivatives are needed to conclude. **Final answer:** D) The test is inconclusive; we need more information.