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.