Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 52

$f(x) $ is increasing for all $x$

Given $$f(x)=x^{3}-2 x^{2}+2 x$$ Since \begin{align*} f'(x)&=3x^{2}-4 x+2 \\ &= 3\left( x^{2}-\frac{4}{3} x+\frac{2}{3} \right)\\ &=3\left( x -\frac{2}{3} \right)^2 +\frac{2}{3} \\ &>\frac{2}{3}>0 \end{align*} Then $f(x) $ is increasing for all $x$