For any twice differentiable function \(f: \mathbb{R} \to \mathbb{R}\), if at some \(x^{*} \in \mathbb{R}\), \(f'(x^{*}) = 0\) and \(f''(x^{*}) > 0\), then the function \(f\) necessarily has a ______ at \(x = x^{*}\).
Note: \(\mathbb{R}\) denotes the set of real numbers.
Answer
(A)
Explanation
This is the second-derivative test. \(f'(x^{*}) = 0\) makes \(x^{*}\) a stationary point, and \(f''(x^{*}) > 0\) means \(f'\) is increasing through \(x^{*}\): \(f' < 0\) just to the left and \(f' > 0\) just to the right. So \(f\) falls and then rises near \(x^{*}\), giving a local minimum; a maximum is impossible.
The test only looks near \(x^{*}\), so a global minimum is not guaranteed: \(f(x) = x^{3} - 3x\) has \(f'(1) = 0\) and \(f''(1) = 6 > 0\), yet \(f(x) \to -\infty\) as \(x \to -\infty\).
Answer: A