MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) \[ f(x)=\sqrt{x}-\frac{1}{9} x, \quad[0,81] \] \[ c= \] Need Help? Read It Watch It Submit Answer

Step 1 ：First, we need to check if the function \(f(x) = \sqrt{x} - \frac{1}{9}x\) is continuous and differentiable on the interval [0, 81] and if \(f(0) = f(81)\).

Step 2 ：The function is continuous on the interval [0, 81] and \(f(0) = f(81)\), but it is not differentiable at x = 0 and x = 81. This is because the derivative of the function is not defined at these points. Therefore, the function does not satisfy all the hypotheses of Rolle's Theorem on the interval [0, 81].

Step 3 ：However, we can still find the values of c in the interval (0, 81) where the derivative of the function is zero, as this might still provide useful information.

Step 4 ：The derivative of the function is zero at x = 20.25. However, since the function is not differentiable at x = 0 and x = 81, this value does not satisfy the conclusion of Rolle's Theorem.

Step 5 ：\(\boxed{\text{Final Answer: The function does not satisfy all the hypotheses of Rolle's Theorem on the interval [0, 81]. Therefore, there are no values of } c \text{ that satisfy the conclusion of Rolle's Theorem.}}\)