Verify that the function satisfies the hypotheses of the...

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=1 / x, \quad[1,3]$

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval [a, b] is computed as (f(b) - f(a))/(b - a). In the context of the Mean Value Theorem, this value represents the slope of the secant line connecting the endpoints (a, f(a)) and (b, f(b)), which is then related to the instantaneous rate of change (derivative) at some point within the interval.

Differentiability

Differentiability on the open interval means that the function has a well-defined derivative at every point within that interval. This is essential for the application of the Mean Value Theorem because the theorem involves equating the derivative at some point c with the average rate of change over the interval. A non-differentiable point would prevent such an equality from being established.

Continuity

Continuity on a closed interval is a key prerequisite for applying the Mean Value Theorem. It means that the function has no breaks, jumps, or holes on the interval, ensuring that it attains every intermediate value between f(a) and f(b). This property is crucial for analyzing the behavior of the function across the entire interval.

Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the instantaneous rate of change (the derivative at c) equals the average rate of change over [a, b]. This theorem provides a formal bridge between the overall change of a function over an interval and its behavior at specific points within that interval.

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