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We are tasked with finding the values of \( m \) and \( b \) such that the given piecewise function \( J(x) \) is continuous at \( x = 3 \) and \( x = 5 \).

The function is defined as:

\[
J(x) = 
\begin{cases}
-2 & \text{if } x \leq 3 \\
mx + b & \text{if } 3 < x < 5 \\
4 & \text{if } x \geq 5
\end{cases}
\]

### Continuity at \( x = 3 \):
For the function to be continuous at \( x = 3 \), the left-hand limit (as \( x \to 3^- \)) and the right-hand limit (as \( x \to 3^+ \)) must be equal to \( J(3) \).

1. Left-hand limit:
\[
\lim_{x \to 3^-} J(x) = -2
\]

2. Right-hand limit (using the middle piece \( mx + b \)):
\[
\lim_{x \to 3^+} J(x) = m(3) + b = 3m + b
\]

For continuity at \( x = 3 \), we need:
\[
-2 = 3m + b
\]

### Continuity at \( x = 5 \):
For the function to be continuous at \( x = 5 \), the left-hand limit (as \( x \to 5^- \)) and the right-hand limit (as \( x \to 5^+ \)) must be equal to \( J(5) \).

1. Left-hand limit (using the middle piece \( mx + b \)):
\[
\lim_{x \to 5^-} J(x) = m(5) + b = 5m + b
\]

2. Right-hand limit:
\[
\lim_{x \to 5^+} J(x) = 4
\]

For continuity at \( x = 5 \), we need:
\[
5m + b = 4
\]

### Solving the system of equations:
We now have the system of two equations:
1. \( 3m + b = -2 \)
2. \( 5m + b = 4 \)

Subtract equation 1 from equation 2:
\[
(5m + b) - (3m + b) = 4 - (-2 a)
\]
\[
2m = 6
\]
\[
m = 3
\]

Substitute \( m = 3 \) into \( 3m + b = -2 \):
\[
3(3) + b = -2
\]
\[
9 + b = -2
\]
\[
b = -11
\]

### Final answer:
The values of \( m \) and \( b \) that make the function continuous at \( x = 3 \) and \( x = 5 \) are:
\[
m = 3, \quad b = -11
\]

Would you like more details or explanations about the solution?

Here are some related questions to explore further:
1. What are the conditions for a function to be continuous at a point?
2. How do you check differentiability of a piecewise function?
3. Can you find the derivative of the given function for \( 3 < x < 5 \)?
4. What happens if a piecewise function is not continuous at a given point?
5. Can a function be continuous but not differentiable at a point?

**Tip:** Continuity at a point requires that the left-hand limit, right-hand limit, and function value all match at that point.

For what values of m and b is J(x) defined as continuous at x = 3 and x = 5?

Discover how to find the values of m and b that make a piecewise function continuous at x = 3 and x = 5. Learn step-by-step how to set up and solve limits, apply continuity, and solve the system of equations. Suitable for advanced high school students.

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