Don't Even Try Guessing
Let f ( x ) = a 0 + a 1 x + . . . . . . . . + a n − 1 x n − 1 + a n x n be a polynomial with non-negative real coefficients, such that f ( 9 ) = 3 and f ( 8 1 ) = 2 7 . Find the maximum possible achievable value of f ( 2 7 ) .
Extra Credit: Find all polynomials having the equality case.
The answer is 9.
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4 solutions
Brilliant question @Satvik Golechha
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Thanks bro, but why d'you take so much trouble to tag like this? You can simply write like:- @shubhendra singh
How did you solve this @Satvik Golechha
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I know two methods. One has already been beautifully illustrated by Tushar, and the other is a direct application of the Cauchy-Schwarz inequality.
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Sorry, but can you explain?
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@Sanjana Nedunchezian – Of course, I will. Let b i 2 = a i , and then apply the Cauchy-Schwarz inequality to get:- f ( 9 ) × f ( 8 1 ) ≥ f ( 2 7 ) . Plug in the values of f ( 9 ) and f ( 8 1 ) to get the answer, that is, 9.
@Sanjana Nedunchezian – You can read about Cauchy-Schwarz inequality here ... The trick is the given condition that the co-efficients are non-negative.....
Yeah...Cauchy Schwarz also produces a nice solution :) !!
Lol I completely guessed it first try
Here we can very easily apply the Cauchy Schwarz inequality to get the required result.
F(9)=9/3=3. F(81)=81/3=27. F(27)=27/3=9
Dude you are too fortunate to get answer with so poor logic!!
Why must that yield the maximum possible achievable value?
All that you have shown is that the answer must be at least 9.
Since the coefficients are positive and f(9) is positive, the derivative is positive for x > 9, which means that f is an increasing function.
Now any polynomial increasing function of any degree will have a radius of curvature causing it to bend downwards, the least being of a straight line. That is, any increasing polynomial function connecting the given two points will show convexity, the least being shown by a straight line.
Thus, finding the equation of the straight line joining the given points, we get y = x/3 implying that max of f(27) = 9.