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#1 Posted 3:24pm 04-11-09 geometrical inequality!!prove that [im]http://codecogs.izyba.com/gif.latex?%5Csqrt%7B%28x_1-x_2%29%5E2+%20%28y_1-y_2%29%5E2%7D%5Cle%20%5Csqrt%7Bx%5E2_1+y%5E2_1%7D+%20%5Csqrt%7Bx%5E2_2+y%5E2_2%7D[/im] for all real numbers [im]http://codecogs.izyba.com/gif.latex?x_1%2Cy_1%2Cx_2%2Cy_2[/im] HINT:[hide]proceed by forming a triangle with suitable coordinates[/hide] |
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#2 Posted 3:31pm 04-11-09 Re: geometrical inequality!!assume triangle with cordinates as origin(0,0) (x1,y1) (x2,y2) we will get this inequality by the fact that sum of lengths of two sides of triangle is always greater than length of third side
"Give me some sunshine,
Give me some rain,
Give me another chance,
I wanna grow up once again"
Edited on 3:33pm 04-11-09 |
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#3 Posted 3:38pm 04-11-09 Re: geometrical inequality!!Deepak bhaiya is correct. but it was for class 9, 10 students. it can also be solved using cauchy schwarz(try it!) |
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#4 Posted 3:39pm 04-11-09 Re: geometrical inequality!!oops.........i din see that.. sorry....and dont call me bhaiya :P.....but seriously how many here on tiit r in class 9-10.......i guess no one :P ....
"Give me some sunshine,
Give me some rain,
Give me another chance,
I wanna grow up once again"
Edited on 3:42pm 04-11-09 |
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#5 Posted 3:48pm 04-11-09 Re: geometrical inequality!!this is a good time to look up Minkowski inequality |
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#6 Posted 09:27am 07-11-09 Re: geometrical inequality!!@theprophet sir thanks for the information. is anybody trying to solve this using cauchy?? |
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#7 Posted 2:29pm 07-11-09 Re: geometrical inequality!!wat is cauchy and minkowski inequality?[7]? i hav never come across dem b4 ever [2]...pls post it if it is helpful....[1]
me : giv me some more time , giv me sum brain , giv me another chance , to write my paper once again ,
examiner : na na na na , na na na na , na na na na na na na na na na !!! |
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#8 Posted 2:45pm 07-11-09 Re: geometrical inequality!!can any body explain minkowski inequality in a simpler way......i saw it in wiki but everything went above my head.....
"lead,follow or get out of the way" |
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#9 Posted 2:46pm 07-11-09 Re: geometrical inequality!!and can anybody post all such iequalities which r useful for jee
"lead,follow or get out of the way" |
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#10 Posted 4:27pm 07-11-09 Re: geometrical inequality!![im]http://codecogs.izyba.com/gif.latex?%28x_1-x_2%29%5E2+%28y_1-y_2%29%5E2%3Dx%5E2_1+x%5E2_2+y_1%5E2+y_2%5E2+2%28x_1.x_2+y_1.y_2%29[/im] ----------------(1) from cauchy schwarz, we know [im]http://codecogs.izyba.com/gif.latex?%28x_1.x_2+y_1.y_2%29%5E2%20%5Cle%20%28x_1%5E2+x_2%5E2%29%28y_1%5E2+y_2%5E2%29[/im] or [im]http://codecogs.izyba.com/gif.latex?%28x_1.x_2+y_1.y_2%29%5Cle%20%5Csqrt%7B%28x_1%5E2+x_2%5E2%29%28y_1%5E2+y_2%5E2%29%7D[/im] from (1) we get [im]http://codecogs.izyba.com/gif.latex?%28x_1-x_2%29%5E2+%20%28y_1-y_2%29%5E2%20%5Cle%20%5Cleft%28%20%5Csqrt%7Bx_1%5E2+x_2%5E2%7D+%5Csqrt%7By_1%5E2+y_2%5E2%7D%5Cright%29%5E2[/im] taking the square root on both sides, we get the desired inequality |
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#11 Posted 5:37pm 14-01-10 Re: geometrical inequality!!squaring we get, -(x[ss]1[/ss]x[ss]2[/ss]+y[ss]1[/ss]y[ss]2[/ss])≤2[[sqrt]x[ss]1[/ss][p]2[/p]+y[ss]1[/ss][p]2[/p][/sqrt] + [sqrt]x[ss]2[/ss][p]2[/p][/sqrt]+y[ss]2[/ss][p]2[/p] again squaring, (x[ss]1[/ss]y[ss]2[/ss]-x[ss]2[/ss]y[ss]1[/ss])[p]2[/p]≥0 which is always true therefore the given equation is also correct. |
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