Use one of the theorems in this section to prove the...

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Key Concepts

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the function's derivative equals the average rate of change over that interval. This theorem is often used to relate the behavior of a function's derivative to the function itself, which in problems like this helps in concluding that a function with zero derivative over an interval must be constant.

Constant Function Theorem

The Constant Function Theorem establishes that if a function's derivative is zero across an interval, then the function must be constant on that interval. In the context of the given problem, by considering the difference between the two functions—whose derivative is zero everywhere—the theorem implies that the difference is a constant, and using a specific initial condition, we demonstrate that this constant is zero, thereby proving that the two functions are identical.

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