Let's analyze the graph and answer the questions.

The function $f(x)$ is graphed, and you are asked to determine the sign (positive or negative) for:

1. $f(0.5)$
2. $f'(0.5)$
3. $f''(0.5)$

Step 1: Evaluate $f(0.5)$

• From the graph, at $x = 0.5$, the function appears to have a negative value (below the x-axis).
• So, $f(0.5)$ is negative.

Step 2: Evaluate $f'(0.5)$

• $f'(x)$ represents the slope of the tangent line to the curve at $x = 0.5$.
• At $x = 0.5$, the function is increasing (the graph is rising), so the slope is positive.
• Thus, $f'(0.5)$ is positive.

Step 3: Evaluate $f''(0.5)$

• $f''(x)$ represents the concavity of the function. If the graph is concave up, $f''(x)$ is positive, and if concave down, $f''(x)$ is negative.
• At $x = 0.5$, the graph appears concave down, meaning $f''(0.5)$ is negative.

Summary:

• $f(0.5)$: Negative
• $f'(0.5)$: Positive
• $f''(0.5)$: Negative

Let me know if you need further clarification or details!

Related Questions:

1. How do you interpret the graph of $f'(x)$ from the original function?
2. What happens to $f'(x)$ when $f(x)$ reaches a maximum or minimum?
3. How does $f''(x)$ affect the shape of the graph of $f(x)$?
4. How do inflection points relate to the second derivative $f''(x)$?
5. Can you describe the behavior of $f(x)$ when both $f'(x)$ and $f''(x)$ are zero?

Tip:

Always check the behavior of the function around the point of interest to confirm concavity and slope!