Monotonic polynomial

What general constraints on the coefficients of a polynomial of degree n do I need to impose to guarantee that this polynomial is strictly increasing on [0,1]?

 

Thank you for your answer, but I am not sure I understand what you mean. Can you explain?

 

Master Level.

 

Yes, it is a Math class which is part of the first year in the master program in economics. Although I am not sure how, I hope this helps.

 

Thank you for your suggestion. Sturm's theorem is something I could use. However, I am not sure how I can use it to find constraints on the coefficients that guarantee the monotonicity of the polynomial over [0,1].

 

I am not sure I understand what you are getting at.

I do not know if it helps or overlap with what you are saying but here is what I tried to do at this point: if P(x) is a polynomial of degree n, then its derivative P'(x) is a polynomial of degree n-1. Therefore, I have tried to parametrize a polynomial of degre n-1 to guarantee that it is strictly greater than 0 for any x between [0,1]. The parametrization I found is

P'(x)=prod(i=1,...n-1){x-1/(1-Bi)} with Bi>0 for any i=1,...n-2 and Bn-1=exp[b*prod(i=1,...n-2){1-Bi}] and b>0 which I believe guarantees that P'(x)>0.

Now however, I am having problems relating the coefficients of P'(x) to the parameters of p(x).

Thank you for your time. I truly appreciate your help.

 

Well, what I am having problems with is to find a formula to relate the (B1,...,Bn-1) in my equation of P'(x) to your (a1,...an).

 

Let me rephrase, what I am having a problem with at this point is to relate the (B1,...,Bn-1) in my definition of P'(x) with the (C1,...,Cn-1) if I write P'(x) in the usual manner

P'(x)=sum(i=1,...,n-1){Ci*x^i}

Then obviously I can easily relate the Ci to your Ai.