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## Lagrange mean value theorem

, 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 $h(x) := \frac{f(b) - f(a)}{b - a}\cdot x + \frac{bf(a) - af(b)}{b - a}$. Then, $h$ is a linear (and hence a continuous and differentiable) function with $h(a) = f(a)$ and $h(b) = f(b)$|| || || || Just plug in and check. Secretly, we obtained $h$ by trying to write the equation of the line joining the points $(a,f(a))$ and $(b,f(b))$.
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| 2 || Define $g = f - h$ on $[a,b]$, i.e., $g(x) := f(x) - h(x)$. || || || ||
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| 3 || $g$ is continuous on $[a,b]$ || Fact (1) || $f$ is continuous on $[a,b]$ || Steps (1), (2) || <toggledisplay>By Step (2), $g = f - h$. By Step (1), $h$ is linear and hence continuous on $[a,b]$. Thus, by Fact (1), $g = f - h$, being the difference of two continuous functions, is continuous on $[a,b]$.</toggledisplay>
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