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Find minimum value of the expression

  1. May 13, 2014 #1


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    1. The problem statement, all variables and given/known data
    Let n be a positive integer. Determine the smallest possible value of $$|p(1)|^2+|p(2)|^2 + .........+ |p(n+3)|^2 $$ over all a monic polynomials p with degree n.

    3. The attempt at a solution

    Let the polynomial be [itex]x^n+c_{n-1} x^{n-1} +.........+ c_1x+c_0 [/itex]

    p(1) = [itex]c_0+c_1+c_2+........+1 [/itex]

    Similarly I can write p(2) and so on, square them and add them together to get a messy expression. But after this, I don't see how to find its minimum value. The final expression is itself difficult to handle. I'm sure I'm missing an easier way to this problem.
  2. jcsd
  3. May 13, 2014 #2


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    Staff: Mentor

    You don't need the full expressions to find derivatives with respect to the coefficients.
  4. May 13, 2014 #3


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    Derivative wrt to which coefficient? There are so many.
  5. May 13, 2014 #4

    Ray Vickson

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    Yo have n variables ##c_0,c_1, \ldots, c_{n-1}## and a function
    [tex] f(c_0,c_2, \ldots, c_{n-1}) = \sum_{k=1}^{n+3} [k^n + c_{n-1} k^{n-1} + \cdots + c_1 k + c_0]^2 [/tex]
    You minimize ##f## by setting all its partial derivatives to zero; that is, by setting up and solving the equations
    [tex] \frac{\partial f}{\partial c_i} = 0, \: i = 0, 1, 2, \ldots, n-1[/tex]
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