# Thread: Prove that g(x) greater than/equal to 0?

1. ## Prove that g(x) greater than/equal to 0?

Let $f(x) = x^2 - 2x$, $x\in \mathbb{R}$ and $g(x) = f(f(x) - 1) + f(5 - f(x))$. Show that $g(x)\geq 0\ \forall\ x\in \mathbb{R}$.

2. Originally Posted by fardeen_gen
Let $f(x) = x^2 - 2x$, $x\in \mathbb{R}$ and $g(x) = f(f(x) - 1) + f(5 - f(x))$. Show that $g(x)\geq 0\ \forall\ x\in \mathbb{R}$.
This is just an algebraic exercise in applying simple identities. I will only show an outline... fill the details

Let $a = x^2 -2x-1$, then $g(x) = a^2 - 2a + (a+4)^2 - 2(a+4) = 2(a+2)^2 = 2(x - 1)^4 \geq 0$