Question

判断f(x)=x^3+4x在（正无穷，负无穷）上的单调性，并用定义证明

Answer 1

f(x)=x^3+4x在(-∞,+∞)递增 任取x1<x2,所以 f(x1)-f(x2)=(x1)^3+4x1-(x2)^3-4x2 =(x1)^3-(x2)^3+4(x1-x2) =(x1-x2)[(x1)^2+x1x2+(x2)^2]+4(x1-x2) =(x1-x2)[(x1)^2+x1x2+(x2)^2+4] =(x1-x2){[(x1+(1/2)x2]^2+(3/4)x^2+4} 因为x1<x2 所以x1-x2<0 因为[(x1+(1/2)x2]^2+(3/4)x^2+4>0 所以f(x1)-f(x2)<0 所以f(x)在R上是递增函数