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. Step 2 a. If g(x) = , then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. Given g(x) = x g (x) = 1 Since g (x) = 1is finite for all values of xin [a,b] g(x)is differentiable on (a,b) .. (1) Since "Every differentiable function is continuous"we get, g(x)is continuous on [a,b] … (2) From (1) and (2) we get, g(x)is continuous on [a,b] and g(x)is differentiable on (a,b) Also g(a) = aand g(b) = b g(a) = / g(b) Thus g(x) = xsatisfies the required condition of generalized mean value theorem. Therefore f(x)and g(x) = xsatisfies the required condition of generalized mean value theorem. So we can apply generalized mean value theorem. Then, there is a point c in (a, b) at which g(b) g(a)= g(c)) Substituting the known values we get, f(b) f(a) f(c) b a = 1 f(b) f(a) f (c) = b a Which is the conclusion in mean value theorem. Hence if g (x) = x, then the Generalized Mean Value Theorem reduces to the Mean Value Theorem.