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.
$f(x)=\ln x, \quad[1,4]$

Key Concepts

Continuity on Closed Intervals

Continuity on a closed interval is a property that guarantees the function does not have any breaks, jumps, or asymptotes within the interval. This smoothness is crucial for the application of the Mean Value Theorem, as it ensures that the function takes on every intermediate value between its endpoints.

Equating Instantaneous and Average Rates of Change

Once the derivative of the function is computed, the next step is to set this derivative equal to the average rate of change of the function over the interval. This process finds the specific point(s) in the open interval where the function behaves 'averagely', fulfilling the statement of the Mean Value Theorem and linking local behavior to global behavior on the interval.

Derivative Computation

Computing the derivative of a function is essential when applying the Mean Value Theorem, as the theorem guarantees that there is a point in the interval where the function’s derivative equals the overall average rate of change. This often involves using differentiation rules to find the derivative and then setting it equal to the ratio of the difference in function values over the interval.

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus that relates the average rate of change of a function over a closed interval to its instantaneous rate of change at some point within the interval. The theorem requires that the function is continuous on the closed interval and differentiable on the open interval, ensuring that there exists at least one point where the derivative equals the function’s average rate of change over the interval.

Differentiability on Open Intervals

Differentiability on an open interval means that the function has a well-defined derivative at every point within that interval. This property ensures that the function’s instantaneous rate of change exists at each point, a necessity for applying the Mean Value Theorem, which relies on equating the derivative at a point to the average rate of change over the interval.

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