4 pts) For what value(s) of the constants 𝑎𝑎 is the function 𝑓𝑓 continuous at 𝑥𝑥 = 0?
𝑓𝑓(𝑥𝑥) = � 𝑎𝑎(𝑥𝑥2 + 2𝑎𝑎) 𝑖𝑖𝑖𝑖 𝑥𝑥 < 0
4 + 𝑎𝑎√𝑥𝑥 + 4 𝑖𝑖𝑖𝑖 𝑥𝑥 ≥ 0

To determine for what values of \( a \) the function \( f(x) \) is continuous at \( x = 0 \), we need to ensure the left-hand limit, the right-hand limit, and the value of the function at \( x = 0 \) are all equal.

The given function is:

\[
f(x) =
\begin{cases} 
a(x^2 + 2a) & \text{if } x < 0 \\
4 + a\sqrt{x} + 4 & \text{if } x \geq 0
\end{cases}
\]

### Step 1: Evaluate \( f(0) \)
From the definition of \( f(x) \) when \( x \geq 0 \), the value of the function at \( x = 0 \) is:
\[
f(0) = 4 + a\sqrt{0} + 4 = 4 + 0 + 4 = 8
\]

So, \( f(0) = 8 \).

### Step 2: Left-hand limit \( \lim_{x \to 0^-} f(x) \)
For \( x < 0 \), we use the expression \( f(x) = a(x^2 + 2a) \). As \( x \to 0^- \), this becomes:
\[
\lim_{x \to 0^-} f(x) = a(0^2 + 2a) = a(2a) = 2a^2
\]

### Step 3: Right-hand limit \( \lim_{x \to 0^+} f(x) \)
For \( x \geq 0 \), we use the expression \( f(x) = 4 + a\sqrt{x} + 4 \). As \( x \to 0^+ \), this becomes:
\[
\lim_{x \to 0^+} f(x) = 4 + a\sqrt{0} + 4 = 8
\]

### Step 4: Continuity condition
For the function to be continuous at \( x = 0 \), the left-hand limit, right-hand limit, and \( f(0) \) must all be equal. Thus, we require:
\[
2a^2 = 8
\]

Solving for \( a \):
\[
a^2 = 4 \quad \Rightarrow \quad a = \pm 2
\]

### Conclusion:
The function \( f(x) \) is continuous at \( x = 0 \) if \( a = 2 \) or \( a = -2 \).

Would you like further details or have any questions? Here are five related questions:

1. How do you verify if a function is continuous at a point?
2. What happens if the limits from the left and right do not match at a point of interest?
3. How do square roots affect continuity conditions?
4. Can you describe why piecewise functions require checking limits at the transition point?
5. How would the problem change if the expression for \( f(x) \) were different for \( x \geq 0 \)?

**Tip**: When solving continuity problems, always check both the left-hand and right-hand limits along with the value of the function at the point.

For what value(s) of the constants a is the function f continuous at x = 0? f(x) = { a(x^2 + 2a) if x < 0, 4 + a√x + 4 if x ≥ 0 }

Learn how to determine the value of the constant 'a' that makes a piecewise function continuous at x = 0. This problem involves evaluating limits from both sides and applying the continuity condition. Suitable for Grades 11-12 students studying calculus or precalculus.

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