The image shows a graph of a function $f$ with a multiple-choice question asking which of the following statements must be true regarding the graph of the derivative $f'(x)$:

The answer options are as follows:

A. $f$ has a relative maximum at $x = 1$. B. $f$ has a relative minimum at $x = 2$. C. $f'(x) < 0$ for $0 < x < 1$. D. $f'(x) > 0$ for $1 < x < 3$.

Let's analyze the graph and options:

1. Option A suggests a relative maximum at $x = 1$. This would mean $f'(1) = 0$ and the slope changes from positive to negative.
2. Option B suggests a relative minimum at $x = 2$, meaning $f'(2) = 0$ and the slope changes from negative to positive.
3. Option C suggests that the slope of the function (or $f'(x)$) is negative between $x = 0$ and $x = 1$.
4. Option D suggests that the slope of the function is positive between $x = 1$ and $x = 3$.

I will now evaluate the graph and confirm which options are correct.Based on the graph and the problem's analysis:

1. Option A: $f$ has a relative maximum at $x = 1$.

   • At $x = 1$, the function appears to reach a peak, meaning that $f'(x)$ would change from positive to negative. This implies that there is a relative maximum at $x = 1$, so Option A is true.

2. Option B: $f$ has a relative minimum at $x = 2$.

   • At $x = 2$, the function seems to have a valley, where $f'(x)$ would change from negative to positive. This suggests that there is a relative minimum at $x = 2$, so Option B is true.

3. Option C: $f'(x) < 0$ for $0 < x < 1$.

   • For values between $x = 0$ and $x = 1$, the function's slope is positive as the curve is increasing. Therefore, $f'(x) > 0$ in this interval, making Option C false.

4. Option D: $f'(x) > 0$ for $1 < x < 3$.

   • For $x$ between 1 and 3, the function first decreases and then starts increasing after $x = 2$. This means that the slope is not entirely positive in this interval, so Option D is false.

Conclusion:

• The correct statements are Option A and Option B.

Would you like any further details or clarifications?

Here are five related questions to explore further:

1. How do you determine the concavity of a function from its graph?
2. What is the relationship between the first and second derivatives of a function?
3. How do inflection points relate to the second derivative?
4. What is the significance of critical points in finding maxima and minima?
5. How can you estimate the value of a derivative from a graph?

Tip: To determine where a function is increasing or decreasing, observe the slope of the tangent lines: increasing when positive, decreasing when negative.