Question

设函数F(x)=|LgX| (x>0) 若a不等于b,并且f(a)=(1/2)f(b)=f((a+b)/2) .求a和b的值.

Answer 1

f(a)=f[(a+b)/2],|lga|=|lg[(a+b)/2]| a不等于b,a不等于(a+b)/2所以lga=-lg[(a+b)/2] a=2/(a+b),a^2+ab=2,|lga|=0.5|lgb|,lgb=2lga或-2lga,所以 b=a^2或b=1/a^2,b=a^2,a^3+a^2-2=0,(a-1)(a^2+2a+2)=0,a=1,b=1不符合题意 b=1/a^2,a^2+1/a=2,a^3-2a-1=0,(a+1)(a^2-a-1)=0,a,b都大于0,所以 a^2-a-1=0,a=(1+√5)/2,b=(3-√5)/2