To analyze the function $f(x) = x^3 - 6x^2 + 3$, we need to find its critical numbers, determine the intervals on which it is increasing or decreasing, and use the First Derivative Test to classify the critical points.

Step 1: Finding Critical Numbers

Critical numbers occur where the derivative $f'(x)$ is zero or undefined.

1. Compute the first derivative $f'(x)$: $f'(x) = \frac{d}{dx}(x^3 - 6x^2 + 3) = 3x^2 - 12x$
2. Set the derivative equal to zero and solve for $x$: $3x^2 - 12x = 0$ Factor out $3x$: $3x(x - 4) = 0$ So the critical numbers are: $x = 0, \quad x = 4$

Step 2: Determining Intervals of Increase and Decrease

To determine where $f(x)$ is increasing or decreasing, we analyze the sign of $f'(x)$ on the intervals determined by the critical points.

1. Choose test points in the intervals $(-\infty, 0)$, $(0, 4)$, and $(4, \infty)$.

   • For $x \in (-\infty, 0)$: $f'(-1) = 3(-1)^2 - 12(-1) = 3 + 12 = 15 \quad (\text{positive})$
   • For $x \in (0, 4)$: $f'(2) = 3(2)^2 - 12(2) = 12 - 24 = -12 \quad (\text{negative})$
   • For $x \in (4, \infty)$: $f'(5) = 3(5)^2 - 12(5) = 75 - 60 = 15 \quad (\text{positive})$

Therefore, $f(x)$ is:

• Increasing on $(-\infty, 0) \cup (4, \infty)$
• Decreasing on $(0, 4)$

Step 3: First Derivative Test for Relative Extrema

Use the First Derivative Test to classify the critical points $x = 0$ and $x = 4$.

1. At $x = 0$:

   • $f'(x)$ changes from positive to negative.
   • Hence, $x = 0$ is a relative maximum.

2. At $x = 4$:

   • $f'(x)$ changes from negative to positive.
   • Hence, $x = 4$ is a relative minimum.

Summary

• Critical numbers: $x = 0, 4$
• Increasing intervals: $(-\infty, 0) \cup (4, \infty)$
• Decreasing intervals: $(0, 4)$
• Relative maxima: $x = 0$
• Relative minima: $x = 4$

Would you like more details or have any questions?

Here are some related questions:

1. How do we find the second derivative of $f(x)$ and use it to confirm concavity?
2. What is the behavior of $f(x)$ as $x$ approaches $\pm\infty$?
3. Can we identify any points of inflection for $f(x)$?
4. How do the relative extrema of $f(x)$ relate to the overall graph of the function?
5. What are the coordinates of the relative extrema points for $f(x)$?
6. How can we sketch the graph of $f(x)$ using the critical points and intervals of increase/decrease?
7. What is the absolute maximum and minimum of $f(x)$ on a closed interval, e.g., $[0, 5]$?
8. How does the graph of $f(x) = x^3 - 6x^2 + 3$ compare to other cubic functions?

Tip: When analyzing a function, always check for critical points and intervals of increase/decrease to understand its behavior thoroughly.