Maximal values of polynomial in Matlab

Given J(x)=(p(x)^2)/q(x) where both p and q are polynomials defined in the workspace by horizontal coefficient arrays, and q(x) > 0 for all values of x.

Create a Matlab script that will compute and display the real and finite value of x for which J(x) is maximized, and also display the maximal value of J achieved at this x value (using only numerical calculations and not symbolic Matlab calculations).

The script should find the maxima using dJ(x)/dx=0 .

I am fairly new working with Matlab and I am unsure on how to approach this problem. I have tried:

P=conv(p,p);
Q=q;
[a,b]=deconv(P,Q) %J(x)
[c,d]=deconv((conv(Q,polyder(P))+conv(P,polyder(Q))),conv(Q,Q)) %dJ(x)/dx
r=roots([c,d])
J=max(abs(r))

But I am not getting the correct answer. For p(x)=2x+5 (i.e. [2 5]) and q(x)=x^4+x^2+2x+2 (i.e. [1 0 1 2 2]), the maximal of J should be approximately 12.9 occurring at x=-0.27. Your help is greatly appreciated.

Answer 1

[David Hill]

p=[2 5];
q=[1 0 1 2 2];
x=-1:.0001:1;
J=@(x)polyval(p,x).^2./polyval(q,x);
[~,idx]=min(abs(diff(J(x))));
Xmax=x(idx);
Jmax=J(Xmax);