Let's analyze and simplify the function $f(x) = 6x^3 + 3x^2 + 3$. Here are some key aspects of the function:

1. Simplification

The function is already simplified as $f(x) = 6x^3 + 3x^2 + 3$.

2. Domain

The domain of the function is all real numbers ($\mathbb{R}$) because there are no restrictions like division by zero or square roots of negative numbers.

3. Derivative (Slope of the function)

To find the derivative $f'(x)$, we differentiate the function with respect to $x$:

$f'(x) = \frac{d}{dx}(6x^3) + \frac{d}{dx}(3x^2) + \frac{d}{dx}(3)$

Calculating each term:

$f'(x) = 18x^2 + 6x$

4. Critical Points

Critical points occur where $f'(x) = 0$:

$18x^2 + 6x = 0$

Factoring:

$6x(3x + 1) = 0$

Thus, the critical points are:

$x = 0 \quad \text{and} \quad x = -\frac{1}{3}$

5. Inflection Points

To find inflection points, let's compute the second derivative $f''(x)$:

$f''(x) = \frac{d}{dx}(18x^2 + 6x) = 36x + 6$

Setting $f''(x) = 0$ to find inflection points:

$36x + 6 = 0 \implies x = -\frac{1}{6}$

Summary

• Derivative: $f'(x) = 18x^2 + 6x$
• Critical Points: $x = 0, -\frac{1}{3}$
• Inflection Point: $x = -\frac{1}{6}$

Would you like details on any specific aspect, or have any questions?

Follow-up Questions

1. How do you find the local maxima and minima using the critical points?
2. What is the concavity of the function around the inflection point?
3. How can the function be graphed using the critical and inflection points?
4. What is the behavior of $f(x)$ as $x \to \infty$ and $x \to -\infty$?
5. How would changes in the coefficients affect the critical and inflection points?

Tip: To analyze the function's behavior, consider evaluating the first and second derivatives at critical points and around inflection points!