- Rolle's Theorem requires a function to be continuous and differentiable on an interval, and to have equal values at its endpoints.
- We evaluated the function \(f(x) = (x - 2)(x + 3)^{10}\) at the endpoints \([-4, 2]\) and found \(f(-4) = -6\) and \(f(2) = 0\).
- Since \(f(-4)
eq f(2)\), the conditions for Rolle's Theorem are not satisfied.
- Therefore, Rolle's Theorem cannot be applied, and we are not guaranteed any points \(c\) in \((-4, 2)\) where \(f'(c) = 0\).
- Final Answer: \(\boxed{\text{Not applicable}}\)
Explanation
- Analysis of Rolle's Theorem Conditions
To determine if Rolle's Theorem applies to the function \(f(x) = (x - 2)(x + 3)^{10}\) on the interval \([-4, 2]\), we must verify three specific conditions:
- The function \(f(x)\) must be continuous on the closed interval \([-4, 2]\).
- The function \(f(x)\) must be differentiable on the open interval \((-4, 2)\).
- The values of the function at the endpoints must be equal, meaning \(f(a) = f(b)\), where \(a = -4\) and \(b = 2\).
Since \(f(x)\) is a polynomial, it is continuous and differentiable for all real numbers, thus satisfying the first two conditions. We proceed to test the third condition by evaluating the function at the boundaries.
- Verifying the Endpoint Values
First, calculate the value of the function at the lower endpoint \(a = -4\):
\(f(-4) = (-4 - 2)(-4 + 3)^{10} = (-6)(-1)^{10} = -6 \times 1 = -6\).
Next, calculate the value of the function at the upper endpoint \(b = 2\):
\(f(2) = (2 - 2)(2 + 3)^{10} = 0 \times 5^{10} = 0\).
Comparing these values, we see that \(f(-4) = -6\) and \(f(2) = 0\). Because \(f(-4)
eq f(2)\), the third condition required for Rolle's Theorem is not satisfied.
- Final Conclusion
Since the condition \(f(a) = f(b)\) is not met, Rolle's Theorem cannot be applied to the function \(f(x) = (x - 2)(x + 3)^{10}\) on the interval \([-4, 2]\). Therefore, it is not guaranteed that there exists a point \(c \in (-4, 2)\) such that \(f'(c) = 0\). The theorem's conclusion does not hold here because the function does not return to its starting value at the end of the interval.
Examples
Rolle's Theorem is a fundamental principle in calculus often used in physics and engineering to analyze systems that return to their original state. For instance, if a rocket is launched from the ground and lands back on the ground, Rolle's Theorem guarantees that at some point during the flight, its vertical velocity must have been zero. In our problem, since the function does not start and end at the same height, the theorem does not apply, demonstrating the importance of checking all conditions before drawing conclusions about a system's behavior.