Saturday, 29 September 2018

continuity - Let f be a continuous and positive function on mathbbR+ such that $lim_{x to infty}



Let f be a continuous and positive function on R+ such that lim.
Prove the equation f(x)=x has at least one solution on \mathbb{R}_{+}.


Answer



If f(0)=0, done. If not g(x)=f(x)-x, g(0)>0 and there exists x>0 such that f(x)/x<1 which implies that g(x)=f(x)-x<0, applies IVT at g in [0,x].


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