91Ó°ÊÓ

The first step in applying Rolle's Theorem is to ensure the function is both continuous and differentiable over the specified interval. With our function, \(f(x) = \sin(3x)\), we are dealing with a sine function, which is known to be continuous and differentiable across all real numbers. This means it does not have any breaks, jumps, or corners that would violate these conditions.

For Rolle's Theorem, the function must be continuous on the closed interval \([a, b]\), and differentiable on the open interval \((a, b)\). In this case, the closed interval is \([\frac{\pi}{2}, \frac{7\pi}{6}]\) and the open interval is \((\frac{\pi}{2}, \frac{7\pi}{6})\). Given the nature of the sine function, we can confidently state that \(f(x)\) meets these requirements in the intervals we've defined, making the first condition for applying Rolle's Theorem satisfied.

Another important condition for Rolle's Theorem is endpoint equality. This means the function must have the same value at both ends of the interval. We check this by calculating \(f(a)\) and \(f(b)\), where \(a = \frac{\pi}{2}\) and \(b = \frac{7\pi}{6}\).

After performing the calculations, we find:

• \(f(\frac{\pi}{2}) = \sin(\frac{3\pi}{2}) = -1\)
• \(f(\frac{7\pi}{6}) = \sin(\frac{7\pi}{2}) = -1\)

Both function values are \(-1\), confirming that \(f(a) = f(b)\). With this verified, we meet the second requirement of Rolle's theorem.

Once continuity, differentiability, and endpoint equality are confirmed, we proceed to find where the derivative is zero within the open interval. This involves differentiating the function \(f(x) = \sin(3x)\). The derivative is \(f'(x) = 3\cos(3x)\).

The next step is to set the derivative equal to zero to find the critical points:

• \(3\cos(3x) = 0\)

Solving this gives \(\cos(3x) = 0\). The general solution to \(\cos(\theta) = 0\) is \(\theta = \frac{(2n+1)\pi}{2}\), for integer values of \(n\).

We then find the specific solution values of \(x\) that lie within the interval \((\frac{\pi}{2}, \frac{7\pi}{6})\). Substituting the general solution into this context helps identify the point(s) where Rolle's Theorem states \(f'(c) = 0\), completing the application of these theorem conditions.