Question

Prove that the function f given by f(x) = x³ − 3x² + 4x is strictly increasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=x^3-3x^2+4x \\ f\text{'}\left(x\right)=3x^2-6x+4 \\ =3\left(x^2-2x\right)+4 \\ =3\left(x^2-2x+1\right)-3+4 \\ =3{\left(x-1\right)}^2+1>0,\forall x\in R \\ \mathrm{Hence}\text{,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is strictly increasing on}R. \end{gathered}$$