Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice-differentiable function, and suppose its second derivative satisfies \( f''(x)>0 \) for all \( x \in \mathbb{R} \). Which of the following statements is/are ALWAYS correct?
Show Hint
- \( f \) has a local minima
- There does not exist \( x \) and \( y \), \( x \neq y \), such that \( f'(x) = f'(y) = 0 \)
- \( f \) has at most one global minimum
- \( f \) has at most one local minimum
The Correct Option is B, C, D
Solution and Explanation
For option (C), because the function is concave up globally, it can only have one global minimum.
For option (B), the second derivative being positive ensures that the first derivative does not change sign more than once, which means the first derivative cannot be zero at two distinct points.
For option (D), because the function is concave up and only has one critical point, it can only have one local minimum.
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