# Integral of a concave function

## Homework Statement

Let $$f:[0,2]\rightarrow[0,\infty)$$ be continuous and non negative. Assume thaqt for any $$x,y\in[0,2]$$ and $$0<\lambda<1 f(\lambda x+(1-\lambda)y)\geq\lambda f(x)+(1-\lambda)f(y)$$. Given that f(1)=1 prove

$$\int_{0}^{2}f(x)dx\geq1$$

## The Attempt at a Solution

I've sat for hours. I have zero inspiration. I need a gentle shove in the right direction please.

## The Attempt at a Solution

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Dick
Homework Helper
Consider the function g(x)=x for x in [0,1] and g(x)=2-x for x in [1,2]. Can you show f(x)>=g(x)?

I think i got it.
Define $$g(x):\mathbb{R}\rightarrow\mathbb{R}$$ . Such that $$g(x)=\begin{cases} x & 0\leq x\leq1\\ 2-x & 1<x<2\end{cases}$$.

Assume that for some $$x\in[0,2] g(x)>f(x)$$ . Then $$g(x)=x>f(x)=f(x1+\left(1-x\right)y)\geq xf(1)+\left(1-x\right)f(y)=g(x)$$ +Something positive (Not precise here with the domain).

Then $$g(x)\geq g(x)+\epsilon$$ a contradiction. Thus we have that for all $$x\in[0,2] f(x)\geq g(x)$$

Then $$\int_{0}^{2}f(x)\geq\int_{0}^{2}f(x)=\int_{0}^{1}x+\int_{1}^{2}2-x=\frac{1}{2}+\left|2x-\frac{x^{2}}{2}\right|_{1}^{2}=\frac{1}{2}+4-2-2+\frac{1}{2}=1$$ Q.E.D.

Dick