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Differentiability is a core concept in calculus, which means that a function has a derivative at every point in its domain. In simpler terms, if a function is differentiable at a point, then it must be smooth without any sharp corners or discontinuities at that point.
- **Derivative**: The derivative of a function at a point gives the slope of the tangent to the curve at that point. It tells us how the function is changing at that specific point.
- **Differentiable Function**: For a function to be differentiable over an interval, it must be differentiable at every single point within that interval.
In the problem, both functions \( f \) and \( g \) are differentiable over the open interval \((a, b)\). This ensures that each of these functions has a smooth slope throughout this interval, making it possible to apply the Mean Value Theorem effectively.

Continuity is a foundational principle in calculus, meaning a function can be drawn without lifting the pencil from the paper. A function is said to be continuous on an interval if there are no breaks, jumps, or gaps over its entire domain.
- **Continuous Function**: A function \( f(x) \) is continuous on a closed interval \([a, b]\) if it is continuous at each point within this interval and has no undefined points.
- **Importance for Mean Value Theorem**: For the Mean Value Theorem to apply, the function must be continuous on the closed interval \([a, b]\).
In the exercise, functions \( f \) and \( g \) are both continuous on the interval \([a, b]\). This continuity ensures that the functions do not have any interruptions, allowing for the application of the Mean Value Theorem.

Function analysis involves examining how a function behaves across its domain. This often includes studying the function's continuity, differentiability, and derivations.
- **Behavior of Functions**: It is crucial to understand both the initial conditions and changes within the function. In our case, the analysis begins by determining that both functions \( f \) and \( g \) are equivalent at point \( a \), so \( f(a) = g(a) \).
- **Comparative Behavior**: Analyzing the differences between two functions over a given range can unveil crucial insights about their relationships. Here, the function \( h(x) = f(x) - g(x) \) helps to simplify and focus this analysis.
In the problem, the analysis through function \( h \) shows the point \( h'(x) < 0 \), thereby methodically leading to the conclusion \( f(b) < g(b) \).

Inequalities play a significant role in calculus as they help in assessing the relative behavior of different functions over certain intervals.
- **Understanding Inequalities**: Inequalities can show how functions compare across an interval. A common situation involves determining if one function is always less than another, as in our case: \( f'(x) < g'(x) \).
- **Analysis via Derivatives**: By understanding the derivatives of functions, one can use inequalities to determine the behavior of functions and their integrals or limits across intervals.
In the given problem, we have \( f'(x) < g'(x) \) which implies that \( h'(x) < 0 \) within \((a, b)\). This inequality ultimately asserts that \( h(b) < 0 \), thus proving \( f(b) < g(b) \). Such results provide a robust foundation for making conclusions in real-world applications where relative positioning of functions is critical.