Question

The minimum value of $f(x)=|x|,\; x\in \mathbb{R}$ is

• A. 0
• B. 1
• C. 2
• D. Does not exist

Hint:

The absolute value function always produces non-negative values, so its minimum value occurs at \(x=0\).

Answer 1

The Correct Option is A

Step 1: Understand the absolute value function.
The absolute value function is defined as \[ |x|= \begin{cases} x, & x\ge 0 \\ -x, & x<0 \end{cases} \] Thus the function always gives a **non-negative value** for every real number \(x\).

Step 2: Examine possible values of the function.
For different values of \(x\): \[ |3|=3 \] \[ |-5|=5 \] \[ |1|=1 \] Hence the value of \( |x| \) is always **greater than or equal to zero**.

Step 3: Identify the smallest value.
The smallest value occurs when \[ x=0 \] because \[ |0|=0 \] No value of \( |x| \) can be less than zero.

Step 4: Conclusion.
Therefore the minimum value of the function \(f(x)=|x|\) is \(0\).
Final Answer: $\boxed{0}$