To answer the questions based on the graph:

Part (a) Find the values of $x$ where $F'(x) = 0$:

The derivative $F'(x) = 0$ at points where the function $F(x)$ has horizontal tangents, meaning the slope of the tangent line is zero. These points correspond to the peaks and troughs on the graph. From the graph, this occurs at points $x = n, r, t$, where the curve has local maxima or minima.

So, the answer is: n, r, t

Part (b) Where is $F'(x) > 0$ (increasing)?:

The function $F'(x) > 0$ when $F(x)$ is increasing, which happens between points where the graph goes upwards (positive slope). Based on the graph:

• $F'(x) > 0$ from $m$ to $n$ (as the curve rises) and from $s$ to $t$ (where it rises again).

Thus, the answer in interval notation is: (m, n), (s, t)

Do you want further details or have any questions?

Here are 5 related questions:

1. How can we determine if a function has a local maximum or minimum using the first derivative?
2. What does $F'(x) < 0$ indicate about the function?
3. How do you find points of inflection using a graph?
4. What is the relationship between the second derivative and concavity?
5. How does the sign of the first derivative relate to the behavior of the original function?

Tip: Remember that the first derivative gives the slope of the tangent to the curve at each point and can help identify intervals of increasing or decreasing behavior!