1. Let's state the problem: You are checking for local maxima and minima of a function and you have found that one critical value's second derivative is zero and the other's second derivative is positive. 2. Recall that the second derivative test is used to classify local maxima and minima for a function $f(x)$ at a critical point $x=c$. According to the test: - If $f''(c) > 0$, then $f$ has a local minimum at $x=c$. - If $f''(c) < 0$, then $f$ has a local maximum at $x=c$. - If $f''(c) = 0$, the test is inconclusive; the critical point could be a max, min, or neither. 3. In your case: - For the critical point where the second derivative is positive: $f''(c) > 0$. Therefore, you have a local minimum at this point. - For the critical point where the second derivative is zero: $f''(c) = 0$. This means the second derivative test does not tell you whether you have a maximum or minimum here. You need to use another method. 4. Alternative methods to classify a critical point when $f''(c) = 0$ include: - Use the first derivative test: examine the sign changes of $f'(x)$ around $c$. - Check higher-order derivatives at $c$ if the function is smooth enough. - Analyze the shape of the function graph or use other problem-specific information. 5. Summary: - Positive second derivative -> local minimum. - Zero second derivative -> inconclusive, further analysis needed. Thus, when the second derivative is zero at a critical point, you cannot conclude the nature of that point from the second derivative test alone.