The function given is:

\[
f(x) = 2x^3 - 2x^2 - 36x + 3
\]

To analyze this function, let's go through the following steps:

### 1. **Find the first derivative \( f'(x) \)**:
This will help us identify critical points (where the slope is zero) and analyze the function's increasing/decreasing behavior.

\[
f'(x) = \frac{d}{dx}(2x^3 - 2x^2 - 36x + 3)
\]

\[
f'(x) = 6x^2 - 4x - 36
\]

### 2. **Find the critical points**:
To find the critical points, set the first derivative equal to zero:

\[
6x^2 - 4x - 36 = 0
\]

Solve for \( x \).

### 3. **Find the second derivative \( f''(x) \)**:
This will help us determine the concavity and identify points of inflection.

\[
f''(x) = \frac{d}{dx}(6x^2 - 4x - 36)
\]

\[
f''(x) = 12x - 4
\]

### 4. **Analyze the behavior of the function**:
Using the critical points and concavity, analyze the function's increasing/decreasing intervals and local maxima/minima.

### 5. **Find the y-intercept**:
To find the y-intercept, set \( x = 0 \):

\[
f(0) = 2(0)^3 - 2(0)^2 - 36(0) + 3 = 3
\]

Thus, the y-intercept is at \( (0, 3) \).

Would you like to proceed with solving for the critical points and concavity, or do you have any questions?

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Here are some related questions:
1. How do you find the roots of a cubic function?
2. What are the critical points, and why are they important?
3. How does the second derivative help determine concavity?
4. What does the y-intercept tell us about the function?
5. How can you graph a cubic function using derivatives?

**Tip:** Always check the sign of the second derivative to understand the concavity of the function.

Learn how to analyze the function f(x) = 2x^3 - 2x^2 - 36x + 3 step-by-step. Explore the calculation of derivatives, finding critical points, determining concavity, and analyzing function behavior. Suitable for advanced high school students and calculus enthusiasts.

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