How can one prove that this polynomial is non-negative?

Question

How one can prove the following inequality?

$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$

I plotted the graph on Wolfram Alpha and found that the inequality seems to hold. I was unable to represent the polynomial as a sum of squares.

It looks quite boring to approximate the derivative to be zero and use some numerical methods to show that values near local minimums proves that the inequality really holds everywhere.

Answer 1

For $x<0$ it's obvious.

But for $x\geq0$ we obtain: $$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74=$$ $$=(x^3-x^2-x+1)(58x^7+16x^6+85x^5+85x^4+207x^3+47x^2)+$$ $$+287x^4-138x^3-103x^2-20x+74>0,$$ where $$287x^4-138x^3-103x^2-20x+74=$$ $$=(16x^2-4x-5)^2+(31x^4-10x^3+x^2)+(40x^2-60x+49)>0.$$

Answer 2

A sum-of-squares decomposition is given by $z^TQz$ where $z = (1,x,x^2,x^3,x^4,x^5)$ and the positive definite matrix $Q$ is

$Q = \begin{bmatrix} 74 & -10 & -38 & 9 & 8 & -30\\ -10 & 20 & 2 & -8 & -22 & 9\\ -38 & 2 & 118 & -28 & -40 & 21\\ 9 & -8 & -28 & 115 & 0 & -32\\ 8 & -22 & -40 & 0 & 75 & -21\\ -30 & 9 & 21 & -32 & -21 & 58 \end{bmatrix}$

Found by solving an integrality constrained sum-of-squares problem (i.e., mixed-integer semidefinite program) in the MATLAB Toolbox YALMIP

sdpvar x
p = 58*x^10-42*x^9+11*x^8+42*x^7+53*x^6-160*x^5+118*x^4+22*x^3-56*x^2-20*x+74;
z = monolist(x,5);
Q = sdpvar(6);
optimize([integer(Q),coefficients(z'*Q*z-p,x)==0,Q>=0],sum(sum(Q)));