Question

求证f(x)=x^3在R（实数）内是增函数..

Answer 1

设a(a>0)为x的增量

则对任意实数x,一定满足x+a>x

f(x+a)-f(x)

=(x+a)^3-x^3

=x^3+3a^2x+3ax^2+a^3-x^3

=3a(x^2+ax+a^2/3)

=3a(x^2+ax+(a/2)^2+a^3/12)

=3a((x+a/2)^2+a^3/12)

=3a(x+(a/2))^2+a^3/4>0

所以对于x+a>x

f(x+a)-f(x)>0

f(x+a)>f(x)

f(x)为增函数