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

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May 23rd 2009, 10:20 PM
fardeen_gen

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 $g(x)\geq 0\ \forall\ x\in \mathbb{R}$ .
May 24th 2009, 02:23 AM
Isomorphism

Quote:

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 (Sleepy)

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

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