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The problem asks whether the function \( f \) is differentiable at \( x = 0 \). To answer this, we need to analyze the differentiability using the definition of the derivative and one-sided limits. Let's examine the problem more closely:

### Problem Analysis:
The graph of \( f \) consists of:
1. A line segment from \( x = -4 \) to \( x = 3 \).
2. A curve that is tangent to the x-axis at \( x = 3 \) and continuous on the interval \( -4 \leq x \leq 6 \).
3. The graph appears to be twice differentiable with \( f(x) > 0 \) for all \( x \).

### Part (a): Is \( f \) differentiable at \( x = 0 \)?
To determine if \( f \) is differentiable at \( x = 0 \), we need to check whether the derivative from both the left-hand side and right-hand side limits exists and are equal.

#### Step 1: Definition of Differentiability:
A function \( f \) is differentiable at \( x = 0 \) if the following limit exists:
\[
\lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}
\]
Additionally, for a piecewise-defined function, we must ensure that:
\[
\lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \quad \text{and} \quad \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h}
\]
are equal.

#### Step 2: Observations from the Graph:
From the provided graph:
- \( f(0) \) is \( 2 \) as shown in the y-axis.
- To find the left-hand limit, we examine the slope of the line segment approaching \( x = 0 \) from the left.
- To find the right-hand limit, we examine the slope of the curve as \( x \) approaches \( 0 \) from the right.

#### Step 3: Slope of the Line Segment:
Based on the graph, the line segment is from \( (-4, 0) \) to \( (0, 2) \). The slope \( m \) of this line segment is:
\[
m = \frac{2 - 0}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}
\]
So, the left-hand limit is:
\[
\lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \frac{1}{2}
\]

#### Step 4: Slope of the Curve (Right-hand Limit):
From the graph, the curve near \( x = 0 \) appears to have a slope of \( 0 \) because it is tangent to the x-axis. Thus:
\[
\lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = 0
\]

#### Step 5: Check Equality:
Since the left-hand limit is \( \frac{1}{2} \) and the right-hand limit is \( 0 \), these limits are not equal. Therefore, \( f \) is **not differentiable** at \( x = 0 \).

### Conclusion:
\( f \) is not differentiable at \( x = 0 \) because the left-hand and right-hand derivatives are not equal. 

Would you like more details or have any other questions?

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Here are five related questions:
1. What is the difference between continuity and differentiability?
2. How can we find if a function is differentiable at a corner point?
3. What is the graphical interpretation of differentiability at a point?
4. Why is it necessary for one-sided derivatives to be equal for differentiability?
5. How would the analysis change if the curve was not tangent to the x-axis at \( x = 3 \)?

**Tip:** Always analyze one-sided derivatives separately when dealing with piecewise or non-linear functions to ensure a proper understanding of differentiability at specific points.

Is f differentiable at x = 0? Use the definition of the derivative with one-sided limits to justify your answer.

Determine if the function f is differentiable at x = 0 using the limit definition of the derivative and one-sided limits. Explore concepts of continuity, slope, and one-sided derivatives to analyze differentiability.

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