80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

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Recall the statement of Cauchy's Mean Value Theorem: If functions \(f\) and \(g\) are continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one \(c \in (a, b)\) such that \[\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}.\]

Verify that \(f(x) = x\) and \(g(x) = x^2\) are continuous on \([-2, 0]\) and differentiable on \((-2, 0)\), which they are since they are polynomials.

Calculate the difference quotient on the right side: \[\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f(0) - f(-2)}{g(0) - g(-2)} = \frac{0 - (-2)}{0^2 - (-2)^2} = \frac{2}{0 - 4} = \frac{2}{-4} = -\frac{1}{2}.\]

Find the derivatives: \[f'(x) = 1, \quad g'(x) = 2x.\]

Set up the equation from Cauchy's Mean Value Theorem: \[\frac{f'(c)}{g'(c)} = \frac{1}{2c} = -\frac{1}{2}.\] Solve for \(c\) in the interval \((-2, 0)\).

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주요 개념

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Cauchy's Mean Value Theorem

Cauchy's Mean Value Theorem generalizes the Mean Value Theorem by relating two functions f and g that are continuous on [a, b] and differentiable on (a, b). It guarantees the existence of a point c in (a, b) where the ratio of their derivatives equals the ratio of their increments: (f'(c)/g'(c)) = (f(b)-f(a))/(g(b)-g(a)).

Differentiability and Continuity Conditions

For Cauchy's Mean Value Theorem to apply, both functions must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). These conditions ensure the existence of the point c and the validity of the theorem's conclusion.

Computing Derivatives and Evaluating at c

To find the value(s) of c, compute the derivatives f'(x) and g'(x), then set their ratio equal to the ratio of the function increments over [a, b]. Solving this equation for c within the interval (a, b) yields the required value(s) satisfying the theorem.

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.a. f(x) = x, g(x) = x², (a, b) = (-2, 0)