Questions: Determine whether Rolle's Theorem applies to the function k(x)=5x/(x^2-4) on the interval [-6,6]. If it applies, find all possible values of c as in the conclusion of the theorem. If the theorem does not apply, state the reason.

Solution Steps

To determine whether Rolle's Theorem applies to the function \( k(x) = \frac{5x}{x^2 - 4} \) on the interval \([-6, 6]\), we need to check the following conditions:

1. The function must be continuous on the closed interval \([-6, 6]\).
2. The function must be differentiable on the open interval \((-6, 6]\).
3. The function values at the endpoints of the interval must be equal, i.e., \( k(-6) = k(6) \).

If all these conditions are satisfied, we then find the derivative of the function and solve for \( c \) in the interval \((-6, 6)\) where the derivative is zero.

The function \( k(x) = \frac{5x}{x^2 - 4} \) has discontinuities at \( x = -2 \) and \( x = 2 \) because the denominator becomes zero at these points. Therefore, \( k(x) \) is not continuous on the interval \([-6, 6]\).

Evaluate the function at the endpoints of the interval: \[ k(-6) = \frac{5(-6)}{(-6)^2 - 4} = \frac{-30}{36 - 4} = \frac{-30}{32} = -\frac{15}{16} \] \[ k(6) = \frac{5(6)}{6^2 - 4} = \frac{30}{36 - 4} = \frac{30}{32} = \frac{15}{16} \]

Since \( k(-6) \neq k(6) \), the function values at the endpoints are not equal.

Since the function \( k(x) \) is not continuous on the interval \([-6, 6]\) and the function values at the endpoints are not equal, Rolle's Theorem does not apply to this function on the given interval.

Final Answer