The given function is $f(x) = 2 + |x - 3|$, and you are asked to find the value of the derivative $f'(x)$ at $x = 3$.

Key points:

• The absolute value function $|x - 3|$ introduces a potential non-differentiability at $x = 3$.
• We will first check if the function is differentiable at $x = 3$.

Step-by-Step Analysis:

1. Piecewise Definition of the Function: The function can be written as:

   x - 3 & \text{if } x \geq 3 \\ 3 - x & \text{if } x < 3 \end{cases}$$

2. Derivative on Either Side of $x = 3$:

   • For $x > 3$, $f(x) = 2 + (x - 3) = x - 1$, and its derivative is $f'(x) = 1$.
   • For $x < 3$, $f(x) = 2 + (3 - x) = 5 - x$, and its derivative is $f'(x) = -1$.

3. At $x = 3$:

   • As we approach from the right ($x \to 3^+$), $f'(x) = 1$.
   • As we approach from the left ($x \to 3^-$), $f'(x) = -1$.
   • Since the left-hand and right-hand derivatives are not equal, the function is not differentiable at $x = 3$.

Conclusion:

The derivative does not exist at $x = 3$.

The correct answer is (E) Nonexistent.

Would you like further explanation on any of the steps?

Here are five related questions:

1. What does it mean for a function to be differentiable at a point?
2. How do you compute the derivative of an absolute value function?
3. What is the significance of the left-hand and right-hand derivatives in determining differentiability?
4. How do piecewise functions affect differentiability and continuity?
5. Can you provide more examples of functions that are not differentiable at a point?

Tip: Always check both continuity and the equality of the left-hand and right-hand derivatives when dealing with absolute value functions at critical points.