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Various derivatives and integrals

Source: Romanian District Olympiad 2024 12.2

March 10, 2024
Integralderivativereal analysis

Problem Statement

Let f:[0,1]→(0,∞)f:[0,1]\to(0,\infty) be a continous function on [0,1][0,1] and let A=∫01f(t)dt.A=\int_0^1 f(t)\mathrm{d}t. [*]Consider the function F:[0,1]→[0,A]F:[0,1]\to[0,A] defined by F(x)=∫0xf(t)dt.F(x)=\int_0^xf(t)\mathrm{d}t.Prove that F(x)F(x) has an inverse function, which is differentiable. [*]Prove that there exists a unique function g:[0,1]→[0,1]g:[0,1]\to[0,1] for which∫0xf(t)dt=∫g(x)1f(t)dt\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}tis satisfied for every x∈[0,1].x\in [0,1]. [*]Prove that there exists c∈[0,1]c\in[0,1] for whichlim⁡x→cg(x)−cx−c=−1,\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,whre gg is the function uniquely determined at b.
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