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Lagrange mean value theorem

5 bytes added, 20:12, 20 October 2011
Proof
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
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| 1 || Consider the function <math>h(x) := \frac{f(b) - f(a)}{b - a}\cdot x + \frac{bf(a) - af(b)}{b - a}</math>. Then, <math>h</math> is a linear (and hence a continuous and differentiable) function with <math>h(a) = f(a)</math> and <math>h(b) = f(b)</math>|| || || || Just plug in and check. Secretly, we obtained <math>h</math> by trying to write the equation of the line joining the points <math>(a,f(a))</math> and <math>(b,f(b))</math>.
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| 2 || Define <math>g = f - h</math> on <math>[a,b]</math>, i.e., <math>g(x) := f(x) - h(x)</math>. || || || ||
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| 3 || <math>g</math> is continuous on <math>[a,b]</math> || Fact (1) || <math>f</math> is continuous on <math>[a,b]</math> || Steps (1), (2) || <toggledisplay>By Step (2), <math>g = f - h</math>. By Step (1), <math>h</math> is linear and hence continuous on <math>[a,b]</math>. Thus, by Fact (1), <math>g = f - h</math>, being the difference of two continuous functions, is continuous on <math>[a,b]</math>.</toggledisplay>
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