Question

The local minimum value of the function \[ f(x) = 3 + |x|, \quad x \in \mathbb{R} \] is:

• A. 1
• B. 2
• C. 3
• D. 0

Hint:

The minimum value of the absolute value function \( |x| \) is 0, and adding a constant does not change the location of the minimum, only its value.

Answer 1

The Correct Option is C

Step 1: The function \( f(x) = 3 + |x| \) consists of a constant 3 and the absolute value function \( |x| \), which is always non-negative. 
Step 2: The absolute value function \( |x| \) reaches its minimum value of 0 at \( x = 0 \). Therefore, the minimum value of \( f(x) \) occurs when \( x = 0 \): \[ f(0) = 3 + |0| = 3. \] Thus, the local minimum value of \( f(x) \) is 3.