To solve this, we first need to analyze the behavior of the function $f(x) = 2x^3 - 15x^2 + 36x - 9$ using calculus. Specifically, we will use the first derivative to find the critical points, which will help determine the intervals where the function is increasing and decreasing, as well as the local minimum and maximum values.

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

To determine where the function is increasing or decreasing, we first calculate the derivative of $f(x)$:

$f'(x) = \frac{d}{dx}(2x^3 - 15x^2 + 36x - 9) = 6x^2 - 30x + 36$

Step 2: Find critical points by setting $f'(x) = 0$

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

$6x^2 - 30x + 36 = 0$

We can simplify by dividing the entire equation by 6:

$x^2 - 5x + 6 = 0$

Now, factor the quadratic equation:

$(x - 2)(x - 3) = 0$

Thus, the critical points are:

$x = 2 \quad \text{and} \quad x = 3$

Step 3: Determine intervals of increase and decrease

To determine where the function is increasing or decreasing, we perform a sign analysis on $f'(x) = 6x^2 - 30x + 36$. We test the sign of $f'(x)$ in the intervals divided by the critical points $x = 2$ and $x = 3$.

• For $x < 2$, say $x = 1$: $f'(1) = 6(1)^2 - 30(1) + 36 = 6 - 30 + 36 = 12 \quad (\text{positive})$

• For $2 < x < 3$, say $x = 2.5$: $f'(2.5) = 6(2.5)^2 - 30(2.5) + 36 = 6(6.25) - 75 + 36 = 37.5 - 75 + 36 = -1.5 \quad (\text{negative})$

• For $x > 3$, say $x = 4$: $f'(4) = 6(4)^2 - 30(4) + 36 = 96 - 120 + 36 = 12 \quad (\text{positive})$

Thus, the function is:

• Increasing on the intervals $(-\infty, 2) \cup (3, \infty)$
• Decreasing on the interval $(2, 3)$

Step 4: Find the local minimum and maximum

• At $x = 2$, the function changes from increasing to decreasing, which indicates a local maximum.
• At $x = 3$, the function changes from decreasing to increasing, which indicates a local minimum.

Now, we find the values of $f(x)$ at these critical points:

• $f(2) = 2(2)^3 - 15(2)^2 + 36(2) - 9 = 16 - 60 + 72 - 9 = 19$
• $f(3) = 2(3)^3 - 15(3)^2 + 36(3) - 9 = 54 - 135 + 108 - 9 = 18$

Final Answers:

(a) The function is increasing on $(-\infty, 2) \cup (3, \infty)$.

(b) The function is decreasing on $(2, 3)$.

(c) The local minimum value is $18$, and the local maximum value is $19$.

Would you like further clarification on any part of this process?

Here are some related questions to deepen your understanding:

1. How would you find the concavity of a function?
2. What does the second derivative tell you about a function's graph?
3. Can you explain why a function changes from increasing to decreasing?
4. How would the process differ for a higher-degree polynomial?
5. How does factoring help simplify solving critical points?

Tip: Always verify your critical points by plugging them back into the original function to ensure the correct identification of maximum and minimum values.