Question

Which of the following functions \( f: \mathbb{R} \to \mathbb{R} \) has/have a local minimum at \( x = 0 \)?

• A. \( f(x) = \sin |x| \)
• B. \( f(x) = \sin x + \frac{x^3}{6} \)
• C. \( f(x) = x^4 + x^2 + 3 \)
• D. \( f(x) = \min \{x - \lfloor x \rfloor, 1 - x + \lfloor x \rfloor \} \)

Hint:

To determine if a function has a local minimum at a point, check for continuity, smoothness, and use the second derivative test when possible.

Answer 1

The Correct Option is A, C, D

Step 1: Analyze option (A).
The function \( f(x) = \sin |x| \) is continuous and differentiable at \( x = 0 \).
However, the function has a sharp corner at \( x = 0 \) due to the absolute value, making it not have a local minimum at \( x = 0 \). Therefore, option (A) is incorrect.

Step 2: Analyze option (B).
The function \( f(x) = \sin x + \frac{x^3}{6} \) is smooth and differentiable. At \( x = 0 \), both the first and second derivatives are zero.
However, the second derivative test fails at \( x = 0 \) because the second derivative is \( 1 \), indicating that \( x = 0 \) is not a local minimum. Therefore, option (B) is incorrect.

Step 3: Analyze option (C).
The function \( f(x) = x^4 + x^2 + 3 \) is a smooth polynomial function. Its first and second derivatives are both continuous, and at \( x = 0 \), the function has a local minimum since the second derivative at \( x = 0 \) is positive (\( f''(0) = 12 \)). Therefore, option (C) is correct.

Step 4: Analyze option (D).
The function \( f(x) = \min \{x - \lfloor x \rfloor, 1 - x + \lfloor x \rfloor \} \) is not continuous at integer points due to the floor function. It does not have a well-defined local minimum at \( x = 0 \), so option (D) is incorrect.

Step 5: Conclusion.
The correct answer is (C), as the function \( f(x) = x^4 + x^2 + 3 \) has a local minimum at \( x = 0 \).