Find the maximum of positive integer $k$ so that for all positive real numbers $x$ we have: $x^6+x^5+x^4+x^2+x+4>kx^3$

Question

Find the maximum of positive integer $k$ so that for all positive real numbers $x$ we have: $x^6+x^5+x^4+x^2+x+4>kx^3$

Since the power of the polynomial on LHS is greater than 3 I have no idea for it!

Answer 1

Assume $k \ge 9$. Then we have $$9=1^6+1^5+1^4+1^2+1+4>k \times 1^3 \ge 9$$ Contradiction. So we have $k<9$.

Note that $$x^6+x^5+x^4+x^3+x^2+x+4=(x^6+4)+(x^5+x)+(x^4+x^2)>8x^3$$ As follows from $\text{AM-GM}$, and because the equality condition can not be satisfied, there is no equality symbol.

So since $8$ is possible, but $k$ is the maximum, we have $k \ge 8$.

So since $k$ is a positive integer that satisfies $$9>k \ge 8$$ we have $k=8$. The answer is $k=8$.

Answer 2

There is a way to solve such problems with arbitrary precision. Let $$P_6(x,k) = x^6+x^5+x^4-kx^3+x^2+x+4,$$ then $$P_5(x,k) = \left(P_6(x,k)\right)'_x = 6x^5+5x^4+4x^3-3kx^2+2x+1.$$ One can looking for value of $k$ in which the system $$P_6(x,k)=0,\quad P_5(x,k)=0$$ is compatible. That means that we can use Euclidean algorythm for calculating of the polynomials $$\gcd(P_6(x,k), P_5(x,k)) = P_4(x,k) = 0,$$ $$\gcd(P_5(x,k), P_4(x,k)) = P_3(x,k) = 0\text{ etc.}$$ One of the roots of $P_0(k)$ is exact value of $k$.

Let us doing that: $$P_4(x,k) = 7x^4-(18k+4)x^3+(3k+24)x^2+28x+143,$$ $$P_3(x,k) = (54k^2+38k-16)x^3 - (9k^2+81k+72)x^2 - (84k+210)x - (429k+233),$$ $$P_2(x,k) = -(81k^4+42k^3-1899k^2+2340k+480)x^2+(54k^3+1582k^2+3576k+2048)x+27k^3+1346k^2+4767k+3448,$$ $$P_1(x,k) = (-118098k^9+677970k^8+11431854k^7+41222868k^6+31860170k^5-65171854k^4-82930534k^3+9323048k^2+21250304k-4558336)x-708588k^9-1384371k^8+30829653k^7+105103098k^6-69349622k^5-367984187k^4-198168467k^3+81825364k^2+43080832k-11876096,$$ $$P_0(k) = -4518872583696k^{22} - 20460450865068k^{21} + 461555336672595k^{20} + 2923486374328344k^{19} - 11846072612516022k^{18}-118906805905404372k^{17} - 36827630500594167k^{16} + 1657522583504454168k^{15} + 3697072310499904796k^{14} - 4396486843108388404k^{13} - 21807742350161679531k^{12}-11026485568012430840k^{11} + 39652086836428096666k^{10} + 62127405370895021940k^9 + 21809194051667172783k^8 - 15873743806067533560k^7 - 11579298157573563216k^6 + 1135988328694170368k^5 + 1895583983245964288k^4 - 15329034111221760k^3 - 129489904818913280k^2 - 513254909542400k + 3163894264627200.$$

The coefficients are particially calculated using the Mathcad package, and $polyroots()$ function is also used, which calculates all the roots of the polynomial by the "accompanying matrix" method.

The real single roots are

$$k\in\{-3.8002, 8.2105\}.$$

Results for the first and the second roots can be illustrated with Wolfram Alpha.