Generalized Mean Value Theorem ?Sup?pose f ? ? and? are functions that are continuous on [a,b] and differentiabl?e o? n (? ,? ).? h?ere ?g? ? )? (?b). Then, there is a point c?in (?a, b) at which f(b) ? f(a) f?(c) g(b) ? g(a) g?c) This result is known as the Generalized (or Cauchy's) Mean Value Theorem. ? ? ? . If ?g(?x)? , then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. b. Suppo ? se f(x) = x ?1,g(x) = ?4x?+2? and [?a, b]= ? [0?,1]. F?ind a value of c? satisfying the Generalized Mean Value Theorem.

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.