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        riddles >> medium >> Quadratic Inequality
        
(Message started by: ThudanBlunder on Oct 27^th, 2008, 12:16pm)

Title: Quadratic Inequality
Post by ThudanBlunder on Oct 27^th, 2008, 12:16pm

If f(x) = ax² + bx + c http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/leqslant.gif 1 for x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/epsilon.gif [0,1] prove that lal + lbl + lcl http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/leqslant.gif 17

Title: Re: Quadratic Inequality
Post by 1337b4k4 on Oct 27^th, 2008, 2:09pm

I'm guessing that |f(x)| needs to be less than 1, not just f(x).

[hide]
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) = 8xhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sup2.gif- 8x + 1[/hide]