Let's analyze the graph and compute the derivatives step by step.

1. $f'(x)$ (first derivative)

The first derivative $f'(x)$ gives the slope of the graph at any point. The graph consists of three linear segments:

• From $x = 0$ to $x = 4$: This is a line with a positive slope.
• From $x = 4$ to $x = 8$: This is a flat (horizontal) line, meaning the slope is 0.
• From $x = 8$ to $x = 10$: This is a line with a positive slope, but steeper than the first segment.

Let's calculate $f'(x)$ at specific points:

$f'(2)$ (slope between $x = 0$ and $x = 4$):

The slope is calculated as: $f'(2) = \frac{4 - 0}{4 - 0} = 1$ So, $f'(2) = 1$.

$f'(6)$ (slope between $x = 4$ and $x = 8$):

This part of the graph is flat, so the slope is zero: $f'(6) = 0$ So, $f'(6) = 0$.

$f'(9)$ (slope between $x = 8$ and $x = 10$):

The slope is: $f'(9) = \frac{8 - 4}{10 - 8} = 2$ So, $f'(9) = 2$.

2. $f''(x)$ (second derivative)

The second derivative represents the rate of change of the slope (i.e., how the slope changes).

$f''(2)$:

Since the slope is constant in the interval $0 \leq x \leq 4$, the second derivative is zero: $f''(2) = 0$

$f''(6)$:

In the interval $4 \leq x \leq 8$, the slope is constant (flat), so the second derivative is also zero: $f''(6) = 0$

3. $f^{(3)}(x)$ (third derivative)

The third derivative is the rate of change of the second derivative. Since the second derivative is zero in all the regions, the third derivative will also be zero at any point.

$f^{(3)}(2)$:

$f^{(3)}(2) = 0$

Final Answers:

• $f'(2) = 1$
• $f'(6) = 0$
• $f'(9) = 2$
• $f''(2) = 0$
• $f''(6) = 0$
• $f^{(3)}(2) = 0$

Would you like more details or have any questions?

Here are some related questions:

1. How is the first derivative related to the slope of a graph?
2. Why is the second derivative zero when the slope is constant?
3. What does it mean when the third derivative is zero?
4. How can we interpret points of inflection using the second derivative?
5. How would the analysis change if there were curves instead of straight lines?

Tip: The first derivative helps you understand the behavior of a function, especially in terms of increasing or decreasing trends and where these trends change.