Can concepts of definite integrals be applied to prove rolles theorom? if so is my proof correct?(adsbygoogle = window.adsbygoogle || []).push({});

Let us consider a function y=g(x). it is continous. the range of y is(-∞,∞).let the derivative of g(x) be f(x)..ie g'(x)=f(x).so the area between the limits a and b is lim(b,a)∫g(x)dx.

So, lim(b,a)∫g(x)dx=f(b)-f(a)+c-----------{1}

We also know area between limits is (b-a)Ʃf{a+(r-1)Δx}/n.--------{2}

Let us take the term Ʃf{a+(r-1)Δx}/n=k-----{3}

We know {2}={3}

So (b-a)Ʃf{a+(r-1)Δx}/n.= lim(b,a)∫g(x)dx=f(b)-f(a)

So, (b-a)*k=f(b)-f(a) [from {3}]

So k=f(b)-f(a)/b-a. Since g(x) is continous, lets say g(c)=k..ie f'(c)=k

hence f'(c)=f(b)-f(a)/b-a

Hence proved

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# New proof of rolles theorom

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