Question

Prove that the function f(x) = x³ − 6x² + 12x − 18 is increasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=x^3-6x^2+12x-18 \\ f\text{'}\left(x\right)=3x^2-12x+12 \\ =3\left(x^2-4x+4\right) \\ =3{\left(x-2\right)}^2\ge 0,\forall x\in R\left[\because 3>0\&{\left(x-2\right)}^2\ge 0\right] \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}R. \end{gathered}$$