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?

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.