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Hi

I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show that the sum of these polynomials also has one peak atmost. Each polynomial is something like $x^3(1-x)^2$ where $0\leq x\leq 1$.

is this information enough.

thanks

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  • $\begingroup$ Do you have a particular reason for asking this? $\endgroup$
    – David Roberts ♦
    Commented Dec 18, 2012 at 4:01
  • $\begingroup$ I am not sure I understand. On the interval $[0,1]$, take the polynomials $p_1(x)=x^2$ and $p_2(x)=(1-x)^2$. Each has maximum $1$, and exactly $1$ peak on this interval, but their sum has two peaks. For a more extreme example, take $p_1(x)=x$ and $p_2(x)=1-x$. Again, both have maximum $1$ and only $1$ peak on $[0,1]$, but their sum is then the constant function. $\endgroup$
    – Eric Naslund
    Commented Dec 18, 2012 at 4:08
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    $\begingroup$ One fact that might be useful is that on a closed, bounded interval on which the sum is strictly concave, it has exactly one maximum. Of course if all the functions are concave on this interval, and at least one is strictly concave, then the sum is strictly concave. $\endgroup$
    – Robert Israel
    Commented Dec 18, 2012 at 4:27

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