Let's analyze the graph and the questions.

The graph is of the function $f(t)$ and you're asked to estimate intervals based on the behavior of the first and second derivatives of this function.

Part (a): Estimating intervals where the derivative $f'(t)$ is positive or negative

1. The derivative $f'(t)$ is positive where the function $f(t)$ is increasing (i.e., where the slope is positive).
2. The derivative $f'(t)$ is negative where the function $f(t)$ is decreasing (i.e., where the slope is negative).

From the graph:

• The function increases from around $t = -2.5$ to $t = 1$, so $f'(t)$ is positive on $-2.5 < t < 1$.
• The function decreases after $t = 1$, meaning $f'(t)$ is negative on $1 < t < 4$.

Part (b): Estimating intervals where the second derivative $f''(t)$ is positive or negative

1. The second derivative $f''(t)$ is positive where the slope of $f(t)$ is increasing (concave up).
2. The second derivative $f''(t)$ is negative where the slope of $f(t)$ is decreasing (concave down).

From the graph:

• The function is concave up (i.e., the slope is increasing) from around $t = -4$ to $t = 0$. So $f''(t) > 0$ on $-4 < t < 0$.
• The function is concave down (i.e., the slope is decreasing) from $t = 0$ to $t = 4$, so $f''(t) < 0$ on $0 < t < 4$.

Let me know if you would like more details or have any questions. Here are some related questions you might consider:

1. How would the intervals change if the function had a local minimum instead of a maximum at $t = 1$?
2. How can you use the sign of the first derivative to classify critical points?
3. How does the concavity of a function relate to points of inflection?
4. What conditions on the second derivative guarantee a local maximum or minimum?
5. How can this analysis be extended to functions with more complicated behavior?

Tip: When analyzing a function, always remember that the first derivative gives you information about increasing/decreasing intervals, while the second derivative tells you about concavity and points of inflection.