https://file.aiquickdraw.com/mathbot/file/1728548843798fd5ojv5d.jpg

Let's solve this step by step.

We are given the piecewise function:

\[
f(x) =
\begin{cases}
2x + 1, & x \leq 0 \\
-x^2 + 2x + 2, & x > 0
\end{cases}
\]

The question is about the differentiability of this function at \( x = 0 \).

### Step 1: Check Continuity at \( x = 0 \)
For a function to be differentiable at a point, it must first be continuous there. So we need to check if \( f(x) \) is continuous at \( x = 0 \).

- For \( x \leq 0 \), \( f(x) = 2x + 1 \).
  \[
  \lim_{x \to 0^-} f(x) = 2(0) + 1 = 1
  \]

- For \( x > 0 \), \( f(x) = -x^2 + 2x + 2 \).
  \[
  \lim_{x \to 0^+} f(x) = -(0)^2 + 2(0) + 2 = 2
  \]

Since \( \lim_{x \to 0^-} f(x) = 1 \) and \( \lim_{x \to 0^+} f(x) = 2 \), the left-hand and right-hand limits are not equal. Hence, \( f(x) \) is **not continuous** at \( x = 0 \).

### Step 2: Differentiability at \( x = 0 \)
Because \( f(x) \) is not continuous at \( x = 0 \), it cannot be differentiable at \( x = 0 \) either. Differentiability implies continuity, so if a function is not continuous at a point, it is automatically not differentiable there.

### Step 3: Conclusion
The function is not differentiable at \( x = 0 \), meaning option **(c)** is correct:  
"**\( f \) is not differentiable at \( x_0 = 0 \)**."

Let me know if you would like more details or have any further questions!

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Here are 5 questions to extend this understanding:
1. What is the definition of differentiability at a point?
2. Can a function be continuous but not differentiable at a point?
3. How do you determine the derivative of a piecewise function?
4. How would you check if the function is continuous for other values of \( x \)?
5. Why does differentiability imply continuity but not vice versa?

**Tip:** Always check continuity before checking differentiability, as a function must be continuous at a point to be differentiable there.

Given the function f(x) = {2x + 1, x ≤ 0 ; -x^2 + 2x + 2, x > 0}, which of the following assumptions is true?

Explore how to determine the differentiability and continuity of a piecewise function at x = 0. This problem involves checking the limits from both sides and understanding the implications of discontinuity on differentiability.

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