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The problem asks for the values of \( a \) and \( b \) such that the function \( f(x) \) is continuous everywhere. The function is defined as follows:

\[
f(x) = \begin{cases} 
a \sin(\pi x) + b & \text{if } x \leq 0, \\
x^2 + a & \text{if } 0 < x \leq 9, \\
b \cos(2 \pi x) + a & \text{if } x > 9.
\end{cases}
\]

### Step 1: Continuity at \( x = 0 \)
For the function to be continuous at \( x = 0 \), the two expressions involving \( x \leq 0 \) and \( 0 < x \leq 9 \) must agree at \( x = 0 \). This means:
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0).
\]
- As \( x \to 0^- \), \( f(x) = a \sin(\pi x) + b \), and since \( \sin(0) = 0 \), we have:
  \[
  \lim_{x \to 0^-} f(x) = b.
  \]
- As \( x \to 0^+ \), \( f(x) = x^2 + a \), and at \( x = 0 \), we have:
  \[
  f(0) = a.
  \]
For continuity at \( x = 0 \), we must have:
\[
b = a.
\]

### Step 2: Continuity at \( x = 9 \)
For the function to be continuous at \( x = 9 \), the two expressions involving \( 0 < x \leq 9 \) and \( x > 9 \) must agree at \( x = 9 \). This means:
\[
\lim_{x \to 9^-} f(x) = \lim_{x \to 9^+} f(x) = f(9).
\]
- As \( x \to 9^- \), \( f(x) = x^2 + a \), and at \( x = 9 \), we have:
  \[
  f(9) = 9^2 + a = 81 + a.
  \]
- As \( x \to 9^+ \), \( f(x) = b \cos(2 \pi x) + a \), and since \( \cos(18\pi) = 1 \), we have:
  \[
  \lim_{x \to 9^+} f(x) = b \cdot 1 + a = b + a.
  \]
For continuity at \( x = 9 \), we must have:
\[
81 + a = b + a.
\]
Simplifying:
\[
b = 81.
\]

### Step 3: Solving for \( a \)
Since from Step 1 we have \( b = a \), substituting \( b = 81 \) into this equation gives:
\[
a = 81.
\]

### Final Answer:
The values of \( a \) and \( b \) that make the function continuous everywhere are:
\[
a = 81, \quad b = 81.
\]

Would you like more details on any part of this solution or have further questions?

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Here are some related questions you might explore:
1. How would the result change if the function had different boundary conditions?
2. What are the general conditions for ensuring continuity in piecewise functions?
3. Can we use derivatives to analyze smoothness (continuity of derivatives) for this function?
4. How does the trigonometric behavior of sine and cosine affect the continuity in such problems?
5. Could the periodicity of sine and cosine functions lead to specific constraints on continuity?

**Tip:** When ensuring continuity of a piecewise function, always equate the boundary limits from both sides of each critical point.

Pour quelles valeurs de a et b la fonction suivante est-elle continue partout?
\[
f(x) = \begin{cases} 
a \sin(\pi x) + b & \text{si } x \leq 0, \\
x^2 + a & \text{si } 0 < x \leq 9, \\
b \cos(2 \pi x) + a & \text{si } x > 9.
\end{cases}
\]

Determine the values of a and b that make the piecewise function f(x) continuous everywhere. The function includes trigonometric expressions (sine and cosine) and polynomial components. Suitable for Grades 10-12 students learning about continuity and limits.

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