Question

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 \(c=\sqrt{a b} .\)

Short answer

The value of \(c\) in the conclusion of the Mean Value Theorem for the function \(f(x) = 1/x\) on any interval [a, b] of positive numbers is \(c = \sqrt{a * b}\).

Step-by-step solution

Recall Mean Value Theorem

The Mean Value Theorem states that if a function f is continuous on a closed interval [a, b], and differentiable on an open interval (a, b), then there exists at least one point in (a, b) such that the derivative at that point is equal to the average rate of change over the interval [a, b]. In mathematical terms, f'(c) = (f(b)-f(a)) / (b-a) for some c in (a,b). We need to find the value of such c.

Apply Mean Value Theorem to f(x)=1/x

With \(a < b\), the function \(f(x) = 1/x\) is continuous on [a,b] and differentiable on (a,b). Thus, by Mean Value Theorem, there exists a c in (a,b) such that \(f'(c) = (f(b)-f(a)) / (b-a)\). So, we have \(-(1/c^2) = (1/b-1/a) / (b-a)\), as \(f'(x) = -1/x^2\).

Solve for c

By cross-multiplying and simplifying, we get: \(-(1/c^2) * (b-a) = 1/b - 1/a\), which implies \(-(b-a) / c^2 = (a - b) / (a * b)\). This simplifies to \(c^2 = a*b\), so \(c = \sqrt{a * b}\), as c is a positive real number (given c in (a, b), a and b are positive).

Key concepts

Geometric Mean

The geometric mean is a type of average, different from the more commonly known arithmetic mean. It is defined as the nth root of the product of n numbers, which for two positive numbers can be expressed as the square root of their product, \(\sqrt{a \times b}\). The geometric mean has special implications in various fields such as finance, statistics, and geometry, where it helps to find the central tendency of multiplicative values. For example, when calculating average rates of growth over time or compound interest rates, the geometric mean provides a more accurate measure than the arithmetic mean.

Continuity and Differentiability

In the context of calculus, we often discuss the properties of continuity and differentiability as fundamental to applying theorems such as the Mean Value Theorem. A function is continuous at a point if it does not have any breaks, jumps, or holes at that point. Moreover, a function is differentiable at a point if it has a definite slope at that point, meaning that its derivative exists. For the Mean Value Theorem to be applicable, the function must be continuous over the closed interval \[a, b\] and differentiable over the open interval \(a, b\). This ensures the existence of at least one point in the interval where the function's instantaneous rate of change matches the function's average rate of change over the entire interval.

Rate of Change in Calculus

The rate of change is a key concept in calculus that describes how a quantity changes with respect to another variable. In real-world terms, it can be understood as speed or velocity—how fast a car's position changes over time, for example. In calculus, we differentiate a function to find its instantaneous rate of change at any given point. This derivative tells us how the function value is expected to change as the independent variable changes by an infinitesimally small amount. It's this fundamental concept that connects with the Mean Value Theorem: the theorem helps to locate a specific point where the function's instantaneous rate of change equals the overall average rate of change between two points.

Derivative Relation to Average Rate

The derivative of a function at a particular point provides the rate of change of the function with respect to its independent variable—essentially capturing the slope of the tangent line at that point. On the other hand, the average rate of change of a function across an interval is the slope of the secant line connecting the end points of the interval on the graph of the function. According to the Mean Value Theorem, for every function that satisfies the theorem's conditions, there exists at least one point within any given interval where the derivative of the function (the slope of the tangent line) is equal to the average rate of change (the slope of the secant line) over that interval. This powerful derivative relationship to the average rate substantiates many real-world phenomena, such as understanding acceleration from velocity.