solve fully also with relative minima and maxima and quadratic function

### Step 1: **First Derivative**
The first derivative of the function is:

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

### Step 2: **Second Derivative**
The second derivative of the function is:

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

### Step 3: **Critical Points**
To find the critical points, we solve the equation \( f'(x) = 0 \):

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

The solutions (critical points) are:

\[
x = \frac{1}{3} - \frac{\sqrt{55}}{3} \quad \text{and} \quad x = \frac{1}{3} + \frac{\sqrt{55}}{3}
\]

### Step 4: **Second Derivative Test (Determine Minima/Maxima)**
- At \( x = \frac{1}{3} - \frac{\sqrt{55}}{3} \), the second derivative \( f''(x) = -4\sqrt{55} \), which is negative. This indicates a **local maximum**.
- At \( x = \frac{1}{3} + \frac{\sqrt{55}}{3} \), the second derivative \( f''(x) = 4\sqrt{55} \), which is positive. This indicates a **local minimum**.

### Step 5: **Values of Function at Critical Points**
The corresponding function values at the critical points are:
- At \( x = \frac{1}{3} - \frac{\sqrt{55}}{3} \), \( f(x) \approx 2\left(\frac{1}{3} - \frac{\sqrt{55}}{3}\right)^3 - 2\left(\frac{1}{3} - \frac{\sqrt{55}}{3}\right)^2 - 36\left(\frac{1}{3} - \frac{\sqrt{55}}{3}\right) + 3 \).
- At \( x = \frac{1}{3} + \frac{\sqrt{55}}{3} \), \( f(x) \approx 2\left(\frac{1}{3} + \frac{\sqrt{55}}{3}\right)^3 - 2\left(\frac{1}{3} + \frac{\sqrt{55}}{3}\right)^2 - 36\left(\frac{1}{3} + \frac{\sqrt{55}}{3}\right) + 3 \).

These correspond to the local maximum and minimum values of the function.

Would you like me to calculate these approximate values explicitly, or do you have any questions about the process?

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Here are some related questions:
1. How can you approximate the roots of a quadratic function if they involve square roots?
2. What is the importance of the second derivative test in determining concavity?
3. How can you sketch the graph of a cubic function using critical points?
4. How do you determine if a critical point is a local minimum or maximum?
5. What are the implications of the function's behavior at infinity for cubic functions?

**Tip:** When analyzing functions with radicals, approximating the values can help visualize their behavior better.

Explore how to solve a quadratic function fully, including finding relative minima and maxima using calculus methods. Learn the step-by-step process of finding critical points and applying the second derivative test. Suitable for advanced high school students and calculus enthusiasts.

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