Is this 'Calculus' or 'Algebra'?
I = ∫ 0 ∞ x 5 e − x d x = ( 2 m 4 + m 3 + 5 m + 9 ) !
Let the product of the real roots (of m ) of the equation above be P .
Given that a + b + c = P , for ( a , b , c ) ∈ R + . Find the Maximum value of:
a + 1 ( 2 a 2 − b 2 − c 2 ) + ( b + c ) 2 + b + 1 ( 2 b 2 − a 2 − c 2 ) + ( c + a ) 2 + c + 1 ( 2 c 2 − b 2 − a 2 ) + ( a + b ) 2
Details and Assumptions :
∙ Give your answer approximately up-to 3 decimal places
∙ The roots of the equation are not necessarily distinct.
This question is part of the set Best of Me
The answer is 1.166.
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1 solution
Sir, but we consider the repeated root every time we solve any problem of this kind, right? Also I have mentioned that a,b,c belong to positive reals, hence there sum can never be negative.
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Well, true, that sort of forces the issue, since a, b, c are all positive reals, so that implies that double negative roots in this case are to be multiplied together. I'll leave it up to you if you want to clarify this for the benefit of would-be solvers.
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As you wish sir,I'll do it right away! BTW what method did you use to evaluate the maximum? Can it be done through any classical inequalities? The only method I'm aware of is Lagrange's multipliers...
The ambiguity with this problem is that in asking for the product of the real roots, is a double root counted twice? That is P = ( − 1 ) ( − 1 ) = 1 ? Otherwise, we get a different answer, if the product of distinct real roots is asked for, in which case P = − 1 .
I think this should be clarified.