(1)由题设知,f(-x)+f(x)=0===>[ax^2+1]/(c-bx)+[ax^2+1]/(c+bx)=0===>[ax^2+1]*[1/(c-bx)+1/(c+bx)]=0===>2c[ax^2+1]/(c-bx)(c+bx)=0===>c=0.又f(1)=2===>(a+1)/b=2===>a=2b-1.又f（2）(4a+1)/2b(8b-3)/(2b)3/2b>1.===>b=1===>a=1.故a=b=1,c=0.f(x)=(x^2+1)/x.(2)由奇偶性,仅考虑x>0时单调性.f(x)=(x^2+1)/x=x+(1/x).易知,在（0,1]上,f(x)递减,在[1,+∞)上,f(x)递增.故由奇函数的性质知,在（-∞,-1]上,f(x)递增,在[-1,0)上,f（x)递减.