Question

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

Answer 1

Given: $f\left(x\right)=x^3-6x^2+12x-18$

To prove: $f\left(x\right)$ is increasing on R.

solution:

$f\left(x\right)=x^3-6x^2+12x-18$

$f^{\prime }\left(x\right)=3x^2-12x+12$

$=3(x^2-4x+4)$

$=3(x-2{)}^2$

$\Rightarrow f^{\prime }\left(x\right)\ge 0$ for all $x\in R$

$\therefore f\left(x\right)$ is increasing for all on $x\in R$.