Let's analyze the graph of the function $y = f(x)$ to answer the questions:

A. Where is the derivative of the function positive?

The derivative is positive where the graph is increasing, i.e., where the slope is positive. Based on the graph:

• The function is increasing between $x = 0$ and $x \approx 1.5$, and also between $x \approx 2.5$ and $x \approx 3.5$.

So, one possible answer is $x = 0.5$.

B. Where is the value of the function positive?

The value of the function is positive when the graph is above the x-axis. Looking at the graph:

• The function is positive for $x$ values between $x = 0$ and $x \approx 3.6$, except at the point where it touches the x-axis.

One possible answer is $x = 1.5$.

C. Where is the derivative of the function largest?

The derivative is largest where the graph has the steepest positive slope, which seems to occur at:

• Around $x \approx 0.5$.

So, the answer is $x = 0.5$.

D. Where is the derivative of the function zero?

The derivative is zero where the slope of the graph is horizontal, i.e., at the local maxima and minima. These points occur at:

• $x \approx 1$ (local maximum),
• $x \approx 2.25$ (local minimum),
• $x \approx 3.25$ (local maximum).

One answer is $x = 1$.

E. Where is the derivative approximately the same as at $x = 1.25$?

To answer this, we need to find where the slope is similar to the slope at $x = 1.25$. At $x = 1.25$, the slope is negative but not steep. Another point where the derivative is similar would be around $x \approx 3.5$.

So, the answer is $x = 3.5$.

Do you have any questions or need further details?

Here are 5 related questions to expand on this topic:

1. How can we estimate the second derivative of a function using its graph?
2. What are the key characteristics of a graph where the derivative changes sign?
3. How can we use a graph to determine concavity and points of inflection?
4. What is the significance of critical points in understanding a function’s behavior?
5. How can we calculate the exact value of a derivative at a given point analytically?

Tip: Always check the slope's steepness and sign (positive or negative) to determine the behavior of the derivative.