# Find minimum value of the expression

1. May 13, 2014

### utkarshakash

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 $x^n+c_{n-1} x^{n-1} +.........+ c_1x+c_0$

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.

2. May 13, 2014

### Staff: Mentor

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

3. May 13, 2014

### utkarshakash

Derivative wrt to which coefficient? There are so many.

4. May 13, 2014

### Ray Vickson

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
$$\frac{\partial f}{\partial c_i} = 0, \: i = 0, 1, 2, \ldots, n-1$$