Let's analyze the given function $f(x) = -3x^3 + 3x^2 - x - 1$ to solve part (a), where we need to find intervals where the function is increasing or decreasing and determine any relative extrema.

Step 1: Find the first derivative $f'(x)$

To determine the intervals of increase or decrease, we first find the derivative of the function:

$f'(x) = \frac{d}{dx} (-3x^3 + 3x^2 - x - 1)$

$f'(x) = -9x^2 + 6x - 1$

Step 2: Find critical points

To find the critical points, set the first derivative equal to zero:

$-9x^2 + 6x - 1 = 0$

Solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = -9$, $b = 6$, and $c = -1$.

$x = \frac{-6 \pm \sqrt{6^2 - 4(-9)(-1)}}{2(-9)}$ $x = \frac{-6 \pm \sqrt{36 - 36}}{-18}$ $x = \frac{-6 \pm 0}{-18}$ $x = \frac{-6}{-18} = \frac{1}{3}$

So, the only critical point is $x = \frac{1}{3}$.

Step 3: Test intervals around the critical point

We now test the intervals around $x = \frac{1}{3}$ to determine where the function is increasing or decreasing.

• For $x < \frac{1}{3}$, choose $x = 0$: $f'(0) = -9(0)^2 + 6(0) - 1 = -1$ Since $f'(0) < 0$, the function is decreasing on $(-\infty, \frac{1}{3})$.

• For $x > \frac{1}{3}$, choose $x = 1$: $f'(1) = -9(1)^2 + 6(1) - 1 = -9 + 6 - 1 = -4$ Since $f'(1) < 0$, the function is also decreasing on $(\frac{1}{3}, \infty)$.

Step 4: Conclusion for part (a)

The function is decreasing on the entire real line, except for the critical point where there is no increase. Therefore, the answer is:

• The function is decreasing on $(-\infty, \infty)$.

Thus, option C: "The function is decreasing on $(-\infty, \infty)$ and increasing nowhere" is the correct choice.

Would you like to proceed with part (b) for concavity and inflection points, or do you have any questions? Here are 5 related questions to practice:

1. What does the first derivative of a function tell you about its graph?
2. How can you determine the concavity of a function using the second derivative?
3. What is the significance of critical points in the graph of a function?
4. How do you classify relative extrema using the first derivative test?
5. Can a function be both increasing and concave down at the same time? Why or why not?

Tip: Always test values around critical points to determine whether the function is increasing or decreasing in each interval.