Question

Show that the function \(f(x)=x^{3}-6x^{2}+12x,\ x\in R\), is an increasing function on \(R\).

Hint:

Differentiate f(x) and show f'(x) is a non-negative perfect square.

Answer 1

Step 1: Computing the derivative:
\(f'(x)=3x^2-12x+12\).

Step 2: Completing the square directly:
\(3x^2-12x+12=3(x^2-4x+4)=3(x-2)^2\), which is always \(\ge 0\) since it's 3 times a square.

Step 3: Concluding monotonicity:
A derivative that is never negative means \(f\) never decreases anywhere on \(R\), i.e. \(f\) is increasing on \(R\).

Final Answer:
\[ \boxed{f \text{ is increasing on } R} \]