It looks like the images you uploaded are related to a calculus problem involving a function $f(x) = 5 - \frac{3x^3}{3}$. In the images, the questions seem to be about finding the derivative of this function, graphing both the function and its derivative, and analyzing the graph to identify certain characteristics of the function and its derivative.

Let's go through the parts of the problem:

Part (a) — Find $f'(x)$

You are given $f(x) = 5 - \frac{3x^3}{3}$. We can simplify it to:

$f(x) = 5 - x^3$

To find the derivative $f'(x)$, we differentiate each term with respect to $x$:

• The derivative of 5 (a constant) is 0.
• The derivative of $-x^3$ is $-3x^2$.

Thus, the derivative is:

$f'(x) = -3x^2$

Part (b) — Graph both $f(x)$ and $f'(x)$

From the images, it looks like the graphs of $f(x) = 5 - x^3$ and its derivative $f'(x) = -3x^2$ are shown. The graph of $f(x)$ should show a cubic curve, while $f'(x)$ is a parabolic curve, as indicated in the graphs.

Part (c) — Use the graph of $f'(x)$ to identify x-values where $f'(x) = 0$

The derivative $f'(x) = -3x^2$ equals zero when $x = 0$. The graph of $f'(x)$ confirms this, as the curve crosses the x-axis at $x = 0$.

Thus, the x-value where $f'(x) = 0$ is $x = 0$.

You also seem to need to identify where $f'(x) > 0$ and $f'(x) < 0$, which corresponds to the regions where the derivative is positive and negative, respectively. From the graph, you can determine:

• $f'(x) > 0$ when $x < 0$ (since the graph of $f'(x)$ is above the x-axis for negative x).
• $f'(x) < 0$ when $x > 0$ (since the graph of $f'(x)$ is below the x-axis for positive x).

Tip

Remember that the sign of the derivative $f'(x)$ tells you about the increasing or decreasing nature of the function $f(x)$. If $f'(x) > 0$, $f(x)$ is increasing; if $f'(x) < 0$, $f(x)$ is decreasing.

Would you like further clarification or help with another part of the problem? Here are some related questions for you to explore:

1. How do you determine the critical points of a function using its derivative?
2. What is the second derivative of $f(x) = 5 - x^3$, and what does it tell you?
3. How do you identify intervals where a function is concave up or concave down?
4. How does the first derivative test help in identifying local maxima and minima?
5. Can you find the inflection points of $f(x) = 5 - x^3$?

Let me know if you need further help!