Find minimum value of the expression

[utkarshakash]

Homework Statement

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.

The Attempt at a Solution

Let the polynomial be $x^n+c_{n-1} x^{n-1} +...+ c_1x+c_0$

p(1) = $c_0+c_1+c_2+...+1$

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.

 

[mfb]

You don't need the full expressions to find derivatives with respect to the coefficients.

 

[utkarshakash]

You don't need the full expressions to find derivatives with respect to the coefficients.

Derivative wrt to which coefficient? There are so many.

 

[Ray Vickson]

Derivative wrt to which coefficient? There are so many.

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