Let's solve each part step by step: --- ### (a) If \( f'(2) = 0 \) and \( f''(2) = 6 \), what can you say about \( f \)? #### **Step 1: Interpret the Conditions** - \( f'(2) = 0 \): The function \( f \) has a **critical point** at \( x = 2 \). - \( f''(2) = 6 \): The second derivative at \( x = 2 \) is **positive**. #### **Step 2: Second Derivative Test** - The second derivative test says: - If \( f''(c) > 0 \) and \( f'(c) = 0 \), then \( f \) has a **local minimum** at \( x = c \). - If \( f''(c) < 0 \) and \( f'(c) = 0 \), then \( f \) has a **local maximum** at \( x = c \). #### **Step 3: Apply the Test** - At \( x = 2 \), \( f'(2) = 0 \) and \( f''(2) = 6 > 0 \). - So, **\( f \) has a local minimum at \( x = 2 \)**. #### **Final Answer for (a):** \[ \boxed{f \text{ has a local minimum at } x = 2.} \] --- ### (b) If \( f'(5) = 0 \) and \( f''(5) = 0 \), what can you say about \( f \)? #### **Step 1: Interpret the Conditions** - \( f'(5) = 0 \): There is a **critical point** at \( x = 5 \). - \( f''(5) = 0 \): The second derivative is zero at the critical point. #### **Step 2: Second Derivative Test** - The second derivative test is **inconclusive** if \( f''(c) = 0 \). - In such cases, you need to look at higher order derivatives or use other methods to determine the nature of the critical point. #### **Step 3: Conclusion** - With the given information, you **cannot determine** whether \( x = 5 \) is a minimum, maximum, or inflection point. #### **Final Answer for (b):** \[ \boxed{\text{The second derivative test is inconclusive at } x = 5; \text{ no conclusion can be drawn about the nature of } f \text{ at } x = 5.} \] --- ### **Summary Table** | Part | Given | Conclusion | |------|-------|------------| | (a) | \( f'(2) = 0, f''(2) = 6 \) | Local minimum at \( x = 2 \) | | (b) | \( f'(5) = 0, f''(5) = 0 \) | Inconclusive; need more information | Let me know if you'd like further explanation!