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Topic: Quadratic Inequality (Read 872 times) |
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ThudnBlunder
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Uberpuzzler
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Quadratic Inequality
« on: Oct 27th, 2008, 12:16pm » |
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If f(x) = ax2 + bx + c  1 for x  [0,1] prove that lal + lbl + lcl 17
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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teekyman
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Re: Quadratic Inequality
« Reply #1 on: Oct 27th, 2008, 2:09pm » |
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I'm guessing that |f(x)| needs to be less than 1, not just f(x).
anyways, let the values of the function at 0,.5,and 1 be x,y,and z. Then we know that
1*c + 0*b + 0*a = x
1*c + .5*b + .25*a = y
1*c + 1*b + 1*a = z
Solving for x,y,and z, we have
c = x
b = -3x + 4y - z
a = 2x - 4y + 2z
thus |c| + |b| + |a| <= |x| + |-3x + 4y - z| + |2x - 4y + 2z| <= 6|x| + 8|y| + 3|z| = 17.
This maximum is attained by f(x) = 8x - 8x + 1
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| « Last Edit: Oct 27th, 2008, 2:16pm by teekyman » |
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