Let $\displaystyle f(x) = x^2 - 2x$, $\displaystyle x\in \mathbb{R}$ and $\displaystyle g(x) = f(f(x) - 1) + f(5 - f(x))$. Show that $\displaystyle g(x)\geq 0\ \forall\ x\in \mathbb{R}$.
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Let $\displaystyle f(x) = x^2 - 2x$, $\displaystyle x\in \mathbb{R}$ and $\displaystyle g(x) = f(f(x) - 1) + f(5 - f(x))$. Show that $\displaystyle g(x)\geq 0\ \forall\ x\in \mathbb{R}$.
This is just an algebraic exercise in applying simple identities. I will only show an outline... fill the details (Sleepy)Quote:
Originally Posted by fardeen_gen
Let $\displaystyle a = x^2 -2x-1$, then $\displaystyle g(x) = a^2 - 2a + (a+4)^2 - 2(a+4) = 2(a+2)^2 = 2(x - 1)^4 \geq 0$
