Means


b. Show that the point guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a,b], where 0 < a < b, is the geometric mean of a and b; that is, c = √ab.

Verified step by step guidance

1. First, recall the Mean Value Theorem (MVT) which states that if a function f(x) is continuous on the 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 f'(c) = (f(b) - f(a)) / (b - a).
2. Verify that f(x) = 1/x is continuous on [a, b] and differentiable on (a, b). Since 0 < a < b, f(x) is continuous and differentiable on these intervals.
3. Calculate f'(x) for f(x) = 1/x. The derivative f'(x) is given by f'(x) = -1/x^2.
4. Apply the MVT: Set f'(c) = (f(b) - f(a)) / (b - a). Substitute f(x) = 1/x to get -1/c^2 = (1/b - 1/a) / (b - a).
5. Solve the equation -1/c^2 = (1/b - 1/a) / (b - a) for c. Simplify the right-hand side and solve for c to show that c = √ab, the geometric mean of a and b.

Key Concepts

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) where the derivative of the function equals the average rate of change over that interval. This theorem is fundamental in understanding the behavior of functions and their derivatives.

Geometric Mean

The geometric mean of two positive numbers a and b is defined as the square root of their product, expressed as √(ab). It is particularly useful in various mathematical contexts, including statistics and finance, as it provides a measure of central tendency that is less affected by extreme values compared to the arithmetic mean.

Derivative

A derivative represents the rate at which a function changes at a given point and is a fundamental concept in calculus. It is defined as the limit of the average rate of change of the function as the interval approaches zero. Understanding derivatives is crucial for applying the Mean Value Theorem, as it helps identify the point where the instantaneous rate of change matches the average rate of change.