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To determine the values of \( a \) and \( b \) that make the function \( f(x) \) continuous over the interval \( [0, \infty) \), let’s analyze the information provided.

The problem states:

\[
f(x) = 
\begin{cases} 
\frac{ax^3 + b}{x^2 - 4} & \text{for } 0 \leq x < 2 \\ 
8 - 6x + 1 & \text{for } x \geq 2 
\end{cases}
\]

To ensure continuity at \( x = 2 \), the left-hand limit (\( \lim_{x \to 2^-} f(x) \)) and the right-hand limit (\( \lim_{x \to 2^+} f(x) \)) must be equal to \( f(2) \).

### Step 1: Right-hand Continuity at \( x = 2 \)
Using the piece defined for \( x \geq 2 \):
\[
f(x) = 8 - 6x + 1
\]
we calculate \( f(2) \):
\[
f(2) = 8 - 6(2) + 1 = 8 - 12 + 1 = -3
\]

### Step 2: Left-hand Limit at \( x = 2 \)
The left-hand side of \( f(x) \) is given by:
\[
f(x) = \frac{ax^3 + b}{x^2 - 4}
\]
For continuity, the limit as \( x \to 2^- \) must be equal to \( f(2) = -3 \). We have:
\[
\lim_{x \to 2^-} \frac{ax^3 + b}{x^2 - 4} = -3
\]

Since \( x^2 - 4 = (x - 2)(x + 2) \), we can substitute \( x = 2 \) only after canceling out the discontinuity in the denominator by making \( ax^3 + b \) divisible by \( x - 2 \).

From the notes on the page, it appears that calculations suggest \( a = 1 \) and \( b = -8 \). However, these values should satisfy:

\[
\lim_{x \to 2^-} \frac{ax^3 + b}{x^2 - 4} = -3
\]

Would you like a further breakdown of these calculations or have any specific questions?

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Here are 5 follow-up questions to deepen understanding:
1. How do we approach continuity problems with piecewise functions generally?
2. Why is it important to check the limits from both sides at the point of continuity?
3. How can we factor and simplify expressions to resolve discontinuities?
4. What other types of points can cause discontinuities in rational functions?
5. How does solving for variables in continuity problems differ from other algebraic solutions?

**Tip:** Always start by ensuring that both side limits exist and match the function's value at the point for continuity in piecewise functions.

Tentukan nilai a dan b agar f(x) kontinu di [0, ∞)

Learn how to determine the values of a and b to ensure the continuity of a piecewise function over the interval [0, ∞). This solution covers essential calculus concepts such as limits and continuity. Ideal for advanced high school math students.

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