Question

Show that f(x) = x³ − 15x² + 75x − 50 is an increasing function for all x ∈ R.

Answer 1

$$\begin{gathered} f\left(x\right)=x^3-15x^2+75x-50 \\ f\text{'}\left(x\right)=3x^2-30x+75 \\ =3\left(x^2-10x+25\right) \\ =3{\left(x-5\right)}^2>0,\forall x\in R\left[\because \mathrm{Square}\mathrm{of}\mathrm{any}\mathrm{function}\mathrm{is}\mathrm{always}\mathrm{greater}\mathrm{than}\mathrm{zero}\right] \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{)is an increasing function for all}x\in R\text{.} \end{gathered}$$