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The problem is asking for which values of \( a \) and \( b \) the function

\[
f(x) = \begin{cases} 
a \sin(\pi x) + b & \text{if } x \leq 1, \\
x^2 + a & \text{if } 1 < x \leq 9, \\
b \cos(2\pi x) + a & \text{if } x > 9 
\end{cases}
\]

is continuous everywhere.

To ensure continuity, we need to check two points where the function changes its definition: at \( x = 1 \) and \( x = 9 \). For the function to be continuous at these points, the left-hand limit and right-hand limit must match the value of the function at those points.

### Step 1: Continuity at \( x = 1 \)

For continuity at \( x = 1 \):
- The value of \( f(x) \) as \( x \to 1^{-} \) is given by \( a \sin(\pi x) + b \).
  \[
  f(1^{-}) = a \sin(\pi \cdot 1) + b = a \sin(\pi) + b = b.
  \]
- The value of \( f(x) \) as \( x \to 1^{+} \) is given by \( x^2 + a \).
  \[
  f(1^{+}) = 1^2 + a = 1 + a.
  \]
- For continuity at \( x = 1 \), we require:
  \[
  f(1^{-}) = f(1^{+}),
  \]
  which gives:
  \[
  b = 1 + a.
  \]

### Step 2: Continuity at \( x = 9 \)

For continuity at \( x = 9 \):
- The value of \( f(x) \) as \( x \to 9^{-} \) is given by \( x^2 + a \).
  \[
  f(9^{-}) = 9^2 + a = 81 + a.
  \]
- The value of \( f(x) \) as \( x \to 9^{+} \) is given by \( b \cos(2\pi x) + a \).
  Since \( \cos(2\pi \cdot 9) = \cos(18\pi) = 1 \), we have:
  \[
  f(9^{+}) = b \cdot 1 + a = b + a.
  \]
- For continuity at \( x = 9 \), we require:
  \[
  f(9^{-}) = f(9^{+}),
  \]
  which gives:
  \[
  81 + a = b + a.
  \]
  Simplifying, we get:
  \[
  b = 81.
  \]

### Step 3: Substituting \( b = 81 \) into \( b = 1 + a \)

From Step 1, we found that \( b = 1 + a \). Substituting \( b = 81 \), we get:
\[
81 = 1 + a,
\]
which simplifies to:
\[
a = 80.
\]

### Final Answer:
- \( a = 80 \)
- \( b = 81 \)

Would you like more details on the solution process, or do you have any further questions?

Here are 5 related questions you might find helpful:
1. How do you prove continuity at a point using limits?
2. What is the significance of left-hand and right-hand limits in determining continuity?
3. Can discontinuity occur at other points, and how can we detect it?
4. How does the sine and cosine function's periodicity affect continuity?
5. How do polynomial functions behave in terms of continuity?

**Tip:** When verifying continuity of piecewise functions, always check the limits at the points where the function's definition changes.

For which values of a and b is the following function continuous everywhere? f(x) = { a*sin(πx) + b if x ≤ 1, x^2 + a if 1 < x ≤ 9, b*cos(2πx) + a if x > 9 }

This problem asks for values of a and b such that a piecewise function involving trigonometric functions (sin(πx) and cos(2πx)) is continuous everywhere. The solution explores the concept of continuity at x = 1 and x = 9 using limits and periodicity of trigonometric functions.

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