Geometric Mean The geometric mean of two positive numbers a...

Geometric Mean The geometric mean of two positive
numbers $a$ and $b$ is $\sqrt{a b}$ . Show that for $f(x)=1 / x$ on any
interval $[a, b]$ of positive numbers, the value of $c$ in the
conclusion of the Mean Value Theorem is $c=\sqrt{a b} .$

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus that states 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 within (a, b) such that the instantaneous rate of change (the derivative) at c is equal to the average rate of change over [a, b]. This theorem provides a formal link between differentiation and the behavior of continuous functions over an interval.

Geometric Mean

The geometric mean of two positive numbers a and b is defined as the square root of their product, ?(ab). It is a type of average that is especially useful in contexts where the numbers are multiplicatively related. This mean is particularly significant in optimization and in problems where products and ratios naturally occur.

Derivative of a Reciprocal Function

For the function f(x) = 1/x, the derivative, which represents the function's instantaneous rate of change, is calculated using standard differentiation rules. The outcome, f'(x) = -1/x², is crucial in applying the Mean Value Theorem to this specific function, as it facilitates the establishment of a relationship between the average and instantaneous rates of change over the interval.

Application to Function on an Interval

When applying these concepts to a function defined on an interval of positive numbers, it is important that the function is continuous and differentiable over the specified range. By verifying these conditions for the reciprocal function, f(x) = 1/x, one can use the Mean Value Theorem to show that there is a specific value c in the interval such that its derivative equals the overall rate of change. In this process, algebraic manipulation ultimately reveals that this c is precisely the geometric mean, ?(ab).

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