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

  1. May 23rd 2009, 10:20 PM #1
    fardeen_gen
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    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}.
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  2. May 24th 2009, 02:23 AM #2
    Isomorphism
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    Quote Originally Posted by fardeen_gen View Post
    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
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