Question

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?

• A. \( f \) has a local minima
• B. There does not exist \( x \) and \( y \), \( x \neq y \), such that \( f'(x) = f'(y) = 0 \)
• C. \( f \) has at most one global minimum
• D. \( f \) has at most one local minimum

Hint:

For twice-differentiable functions with \( f''(x)>0 \), they are concave up everywhere, and thus, they can have only one global and one local minimum, and the two coincide.

Answer 1

The Correct Option is B, C, D

Since \( f''(x) > 0 \) for all \( x \in \mathbb{R} \), the function is concave up everywhere.

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.