Question

The minimum of $f(x)=|x+2|$, $x\in\mathbb{R}$ occurs at ________.

• A. $x=0$
• B. $x=2$
• C. $x=1$
• D. $x=-2$
• E. $x=-1$

Hint:

The vertex of $|x-h|$ is always at $x=h$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
We need to find the value of \(x\) for which the function \(f(x) = |x+2|\) attains its minimum value. The absolute value function \(|y|\) represents the distance of \(y\) from zero. Its value is always non-negative.
Step 2: Key Formula or Approach
The function \(f(x) = |g(x)|\) will have a minimum value of 0. This minimum occurs when the argument of the absolute value is zero, i.e., when \(g(x) = 0\).
Step 3: Detailed Explanation
1. Analyze the function \(f(x) = |x+2|\).
The output of the absolute value function is always greater than or equal to zero.
\[ |x+2| \ge 0 \] 2. Find the minimum value.
The smallest possible value that \(f(x)\) can take is 0. This is the minimum value of the function.
3. Find the value of x where the minimum occurs.
The minimum value is achieved when the expression inside the absolute value is equal to 0.
\[ x+2 = 0 \] Solving for \(x\), we get:
\[ x = -2 \] At \(x = -2\), the function value is \(f(-2) = |-2 + 2| = |0| = 0\). For any other value of \(x\), \(x+2\) will be non-zero, and \(|x+2|\) will be strictly positive. For example, if \(x=-1\), \(f(-1) = |-1+2| = 1>0\).
Therefore, the minimum occurs at \(x=-2\).
Step 4: Final Answer
The minimum of the function occurs at x = -2.