Let's analyze the problem.

We are tasked with determining $f'(0)$, which is the slope of the tangent $T$ to the curve $C_f$ at $x = 0$. Here's the step-by-step process:

1. Identify the tangent $T$: From the graph, the tangent $T$ passes through the point where $x = 0$. This tangent is clearly not horizontal; instead, it has a visible negative slope.

2. Determine the slope visually: The tangent $T$ crosses the y-axis at $y = 1$ and appears to go down 2 units vertically for every 3 units it moves horizontally to the right. Thus, the slope of the tangent $T$ is: $f'(0) = -\frac{2}{3}.$

The correct answer is therefore: $f'(0) = -\frac{2}{3}.$

Let me know if you would like a deeper explanation or have any specific questions!

Here are 5 related questions to expand on this concept:

1. What is the definition of the derivative at a point in terms of limits?
2. How can you interpret the slope of a tangent line geometrically?
3. What is the significance of horizontal tangents in a function's graph?
4. How would you calculate the derivative of a function algebraically at $x = 0$?
5. What does the sign of the derivative (positive/negative) tell you about the behavior of the function?

Tip: Always relate the slope of a tangent to real-world interpretations, like rates of change, to strengthen your understanding.