MathDB

4

Part of 2006 Vietnam National Olympiad

Problems(1)

Vmo 2006 a4

Source:

2/28/2006
Given is the function f(x)=−x+(x+a)(x+b)f(x)=-x+\sqrt{(x+a)(x+b)}, where aa, bb are distinct given positive real numbers. Prove that for all real numbers s∈(0,1)s\in (0,1) there exist only one positive real number α\alpha such that f(α)=as+bs2. f(\alpha)=\sqrt {\frac{a^s+b^s}{2}} .
functioncalculusderivativelimitalgebra unsolvedalgebra