Find minimum value of the expression

In summary, the problem is to determine the smallest possible value of the sum of the squares of a monic polynomial's values at consecutive positive integers, with a given degree, and the solution involves finding the minimum of a function with n variables by setting its partial derivatives to zero.
  • #1
utkarshakash
Gold Member
854
13

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 [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.
 
Physics news on Phys.org
  • #2
You don't need the full expressions to find derivatives with respect to the coefficients.
 
  • #3
mfb said:
You don't need the full expressions to find derivatives with respect to the coefficients.

Derivative wrt to which coefficient? There are so many.
 
  • #4
utkarshakash said:
Derivative wrt to which coefficient? There are so many.

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]
 

FAQ: Find minimum value of the expression

1. What is an expression?

An expression is a mathematical phrase that contains numbers, variables, and operators. It can be evaluated to produce a numerical result.

2. How do you find the minimum value of an expression?

To find the minimum value of an expression, you can use techniques such as substitution, differentiation, or graphing. These methods help to identify the critical points where the expression is at its minimum value.

3. Can an expression have multiple minimum values?

Yes, an expression can have multiple minimum values. This occurs when the expression has multiple critical points that are all local minima.

4. What is the difference between local and global minimum value?

A local minimum value is the smallest value of an expression within a specific interval or range. A global minimum value is the smallest value of an expression among all possible values, regardless of the interval or range.

5. Can a minimum value be negative?

Yes, a minimum value can be negative. This occurs when the expression has negative coefficients or variables that result in a negative value at the minimum point.

Back
Top