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We are given a piecewise function:

\[
f(x) = 
\begin{cases} 
6\sin(x) + 2 & \text{if } x \leq \frac{5\pi}{6} \\
mx + 2 & \text{if } x \geq \frac{5\pi}{6}
\end{cases}
\]

We are tasked with finding the value of \( m \) that makes \( f(x) \) continuous everywhere, particularly at \( x = \frac{5\pi}{6} \).

### Steps to solve:

To ensure the continuity of \( f(x) \), the two pieces of the function must be equal at the point where they meet, i.e., at \( x = \frac{5\pi}{6} \).

1. **Condition for Continuity**:  
   \[
   \lim_{x \to \left(\frac{5\pi}{6}\right)^-} f(x) = \lim_{x \to \left(\frac{5\pi}{6}\right)^+} f(x) = f\left(\frac{5\pi}{6}\right)
   \]

   This implies

What must the value of m be so that the function f(x) is continuous everywhere?

Learn how to find the value of m that makes a piecewise function continuous at a given point. This problem involves trigonometric and algebraic concepts, such as sine functions and continuity conditions, suitable for high school students in Grades 11-12.

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