More from Dr.Vijay S

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Calculus — Il, , Cauchy’s Mean Value Theorem, , Dr.Vijay S, Assistant Professor of Mathematics,, , Government Science College,, Hassan-573 201

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Cauchy's Mean Value theorem, , Statement: Jf f:[a,b] > R and g:|[a,b| > Rare such that, , (i) f, g are continuous on[a, b|, , (ii) f, g are derivable on (a,b), (iii) g' (x) # 0, for any x € (a,b), , FO _fH-f@, gc) g(b)- g(a), , , , then there exists c € (a,b) such that

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Proof: Let f,g : [a,b] > R be such that, (i) f, g are continuous on[a, b], , (ii) f, g are derivable on (a,b), , (iii) g'(x) # 0, for all x € (a,b), , Define (x) = f(x) +K g(x),, , where K € Rchoosen such that $(a) = $(b)., , Now (a) = o(b), , = f@+Kg@ =f@)+K 9g), , => K(g(a)-g(b)) = fb) -f@, , _fO)-F@ 0. va, , = =, g(a)-g(b) g(b)-g(a), , — [f@=860) —, , (), , , , Since g(a) = g(b)contradicts, , to the fact that g'(x) #0

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As f (x) is continuous in[ a,b] and Kg(x) is continuous in [a,b |., => (x) is continuous in [ a, b ]., As f (x) is derivable in (a,b) and Kg(x)isderivablein (a,b),, => p(x) is derivable in (a,b)., Also @ (a) = (bd)., , Hence @ (x) satisfies all conditions of Rolle’s theorem., , Therefore there exist at least one c € (a,b ) such that, p'(c) =0, p(x) = f(x) + K g(), “ P'(x) =f’) +K g(x), , “@'(c)=f'(c)+K g'(c)=0 (Because f'(c) = 0)

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« f'(c) = -K g'(c), , f'(c), ” a 2, g'(c) i?), , , , By equation (1) we have, , _ _ f)-f(@), g(b)—9(a), , , , Substitute in equation (2), , fo _ f®%)-—@, g'(c) g(b)-g(a), , Hence the theorem.