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#1 Posted 7:48pm 24-03-09 Not for JEELike the title says, this is not JEE stuff, but if you have been following some of my recent posts, this may be worthwhile doing: [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20f%28x%29%20%5C%20dx%20%3D%20%5Cint_0%5E1%20x%20f%28x%29%20%5C%20dx%20%3D%201[/im] Prove that [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20f%5E2%28x%29%20%5C%20dx%20%5Cge%204[/im] |
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#2 Posted 08:36am 25-03-09 Re: Not for JEENo bravehearts doing this one? |
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#3 Posted 10:57am 25-03-09 Re: Not for JEEany thing said about f(x)? |
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#4 Posted 11:05am 25-03-09 Re: Not for JEENo, the problem statement is complete. |
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#5 Posted 2:19pm 25-03-09 Re: Not for JEEWe have, for a real [im]http://codecogs.izyba.com/gif.latex?%5Clambda[/im], [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1f%28x%29%28x+%5Clambda%29%5C%20%5Cmathrm%7Bd%7Dx%20%3D1+%5Clambda[/im] Therefore, by Cauchy-Schwarz, we get [im]http://codecogs.izyba.com/gif.latex?%281+%5Clambda%29%5E2[/im] [im]http://codecogs.izyba.com/gif.latex?%3D%5Cleft%28%5Cint_0%5E1f%28x%29%28x+%5Clambda%29%5C%20%5Cmathrm%7Bd%7Dx%5Cright%29%5E2[/im] [im]http://codecogs.izyba.com/gif.latex?%5Cle%20%5Cint_0%5E1f%5E2%28x%29%5C%20%5Cmathrm%7Bd%7Dx%5C%2C%5Cint_0%5E1%28x+%5Clambda%29%5E2%5C%20%5Cmathrm%7Bd%7Dx[/im] [im]http://codecogs.izyba.com/gif.latex?%3D%20%5Cdfrac%7B3%5Clambda%5E2%20+%203%5Clambda%20+1%7D%7B3%7D%5Cint_0%5E1f%5E2%28x%29%5C%20%5Cmathrm%7Bd%7Dx[/im] Hence, we have [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1f%5E2%28x%29%5C%20%5Cmathrm%7Bd%7Dx%20%5Cgeq%20%5Cdfrac%7B3%281+%5Clambda%29%5E2%7D%7B3%5Clambda%5E2+3%5Clambda+1%7D[/im] Since this inequality must hold for all real [im]http://codecogs.izyba.com/gif.latex?%5Clambda[/im], we must have [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1f%5E2%28x%29%5C%20%5Cmathrm%7Bd%7Dx%20%5Cgeq%20%5Cmax_%5Clambda%5Cleft%28%5Cdfrac%7B3%281+%5Clambda%29%5E2%7D%7B3%5Clambda%5E2+3%5Clambda+1%7D%5Cright%29%3D4[/im] with equality at [im]http://codecogs.izyba.com/gif.latex?%5Clambda%20%3D%20-%5Cdfrac%7B1%7D%7B3%7D[/im] and hence for [im]http://codecogs.izyba.com/gif.latex?f%28x%29%3D6%5Cleft%28x-%5Cdfrac%7B1%7D%7B3%7D%5Cright%29[/im] |
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#6 Posted 2:22pm 25-03-09 Re: Not for JEEIt was a nice one and took me a while to figure it out. However, I enjoyed solving it. :) |
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#7 Posted 2:26pm 25-03-09 Re: Not for JEEYeah, the Schwarz Bunyakovsky Inequality is called into play here: I'll write out my solution, which is essentially the same as above, but is a little more straightforward: [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20f%28x%29%20%281+x%29%20%5C%20dx%20%3D%202%20%5CRightarrow%20%5Cint_0%5E1%20f%5E2%28x%29%20%5C%20dx%20%5Cint_0%5E1%20%281+x%29%5E2%20%5C%20dx%20%5Cge%20%5Cleft%28%5Cint_0%5E1%20f%28x%29%20%281+x%29%20%5C%20dx%20%5Cright%29%5E2[/im] But [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20%281+x%29%5E2%20%5C%20dx%20%3D%201[/im] Hence the inequality follows |
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#8 Posted 2:47pm 25-03-09 Re: Not for JEEWell, I first did what you did. However, for equality we require f(x) to be proportional to (1+x). So if we took f(x)=k(1+x), then the condition [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20f%28x%29%5C%20%5Cmathrm%7Bd%7Dx%3D1[/im] give us [im]http://codecogs.izyba.com/gif.latex?k%20%3D%20%5Cdfrac%7B2%7D%7B3%7D[/im]. But with this k, our function at equality becomes [im]http://codecogs.izyba.com/gif.latex?f%28x%29%20%3D%20%5Cdfrac%7B2%7D%7B3%7D%281+x%29[/im] But with this function, the second condition [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20x%20f%28x%29%5C%20%5Cmathrm%7Bd%7Dx%3D1[/im] does NOT hold. Further, in this case [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%20f%5E2%28x%29%5C%20%5Cmathrm%7Bd%7Dx%3D%5Cdfrac%7B28%7D%7B27%7D%5Cneq%204[/im] So, your solution is not entirely correct. |
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#9 Posted 3:13pm 25-03-09 Re: Not for JEEHmm. I didnt verify that. |
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#10 Posted 3:59pm 25-03-09 Re: Not for JEEMoreover, how did you get [im]http://codecogs.izyba.com/gif.latex?%5Cint_0%5E1%281+x%29%5E2%5C%20%5Cmathrm%7Bd%7Dx%3D1[/im] ? |
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#11 Posted 4:20pm 25-03-09 Re: Not for JEEthe problem, as usual, was: 1. i did this in my head 2. i did not bother to check the solution at the back :D Apologies and thank you kaymant sir for the correct solution. Otherwise the students would have been misled |
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