question_answer

Prove that the function given by \[f(x)={{x}^{3}}-3{{x}^{2}}+3x-100\] is increasing in R.

Answer:

We have, \[f(x)={{x}^{3}}-3{{x}^{2}}+3x-100\] On differentiating both sides w.r.t. x, we get \[f'(x)=3{{x}^{2}}-6x+3=3({{x}^{2}}-2x\,+1)\]             \[={{(x\,-1)}^{2}}\ge 0,\] for real values of x.                          \[\therefore \]f(x) is increasing in R .                 Hence proved.