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Generalized Mean Value Theorem Suppose f and are functions
Chapter 3, Problem 39AE(choose chapter or problem)
Suppose f and g are functions that are continuous on [a, b] and differentiable on (a, b), where \(g(a) \neq g(b)\). Then, there is a point c in (a, b) at which
\(\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}\).
This result is known as the Generalized (or Cauchy's) Mean Value Theorem.
a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem.
b. Suppose \(f(x)=x^{2}-1\), g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.
Questions & Answers
QUESTION:
Suppose f and g are functions that are continuous on [a, b] and differentiable on (a, b), where \(g(a) \neq g(b)\). Then, there is a point c in (a, b) at which
\(\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}\).
This result is known as the Generalized (or Cauchy's) Mean Value Theorem.
a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem.
b. Suppose \(f(x)=x^{2}-1\), g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.
ANSWER:Solution 39AE Step 1 Given: Generalized Mean Value Theorem Su ppose f and are functions that are continuous on [a,b]and differentiab le on (a ). w here a) (b). Then, there is a point c in (a, b) at which f(b) f(a) f(c) g(b) g(a)= gc) In this problem we have to show if ( ) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem and we have to find c satisfying the Generalized Mean Value Theorem with the given functions.