Question

The derivative of \(f(x) = 3|2 + x|\) at the point \(x_0 = -3\) is

• A. 3
• B. -3
• C. 0
• D. None of the above

Hint:

\(\frac{d}{dx}|x| = \frac{x}{|x|}\) for \(x \neq 0\).

Answer 1

The Correct Option is B

The problem requires finding the derivative of the function \(f(x) = 3|2 + x|\) at the point \(x_0 = -3\). Since the function involves an absolute value, we must consider the piecewise definition of the function and its derivative. 

1. **Understanding the Function \(f(x)\):**

• For the function \(f(x) = 3|2 + x|\), let's consider two cases based on the expression inside the absolute value:

1. Case 1: \(2 + x \geq 0\) or \(x \geq -2\). Here, \( |2 + x| = 2 + x\).
2. Case 2: \(2 + x < 0\) or \(x < -2\). Here, \( |2 + x| = -(2 + x) = -2 - x\).

2. **Finding the Derivative \(f'(x)\):**

• For \(x \geq -2\), the function \(f(x) = 3(2 + x)\).
• Derivative: \(f'(x) = 3 \cdot 1 = 3\).
• For \(x < -2\), the function \(f(x) = 3(-2 - x)\).
• Derivative: \(f'(x) = 3(-1) = -3\).

3. **Evaluate at \(x_0 = -3\):**

• Since \(-3 < -2\), the relevant case is when \(x < -2\), where the derivative \(f'(x) = -3\).

Thus, the derivative of \(f(x)\) at \(x_0 = -3\) is \(-3\).

Therefore, the correct answer is -3.