Which of the following is/are correct? 1. f^{\prime}(0)=0 2. f^{\prime\prime}(0) \lt 0 Select the correct answer using the code given below :
Consider the following for the next two (02) items that follow : A differentiable function f(x) has a local <strong>MAXIMUM</strong> at x=0. Let y=2f(x)+ax-b.
- A. 1 only
- B. 2 only
- C. Both 1 and 2 ✓
- D. Neither 1 nor 2
Correct Answer: C. Both 1 and 2
Explanation
Since the function f(x) is differentiable and has a local maximum at x=0, its first derivative at that point must be zero, so f'(0)=0. Furthermore, by the second derivative test, the second derivative must be negative at a local maximum, meaning f''(0) \lt 0. Thus, both statements are mathematically valid conditions.
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