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Prove that f=ax+b

Source:

October 31, 2019

functioncalculusderivativereal analysislagrangeMVTConvexity

Problem Statement

Let be two real numbers

a<b

and a differentiable function

f:[a,b]\longrightarrow\mathbb{R}

that has a bounded derivative. Show that if

\frac{f(b)-f(a)}{b-a}

is equal to the global supremum or infimum of

f',

then

f

is polynomial with degree

1.

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