of $7^{th}$ degree. The third derivate of the polynomial $Q_7(x)$ is the polynomial of $4^{th}$ degree $$\label{Ineq_91}
\begin{array}{rcl}
Q_7^{'''}(x)\!\!&\!\!\!=\!\!\!&\!\!210(-47520\pi^2+475200)x^4+120(1024\pi^5+23760\pi^3\\ [1.0 ex]
\!\!&\!\!\! \!\!\!&\!\!-285120\pi)x^3+60(498960\pi^2\!-\!4989600)x^2+24(-6963\pi^5\\ [1.0 ex]
\!\!&\!\!\! \!\!\!&\!\!-249480\pi^3+2993760\pi)x-14968800\pi^2+149688000.
\end{array}$$ Using MATLAB software we can determine the real numerical factorization of the polynomial $$\label{Ineq_92}
Q_7^{'''}(x)=\alpha(x- x_1)(x- x_2)(x- x_3)(x- x_4),$$ with values $\alpha= 1.301\ldots 10^6$, $ x_1=-14.400\ldots $, $ x_2=-0.776\ldots $, $ x_3=0.174\ldots $, $ x_4=0.768\ldots\;$. The polynomial $Q_7^{'''}(x)$ has exactly four simple real roots with a symbolic radical representation and the corresponding numerical values $x_1, x_2, x_3,  x_4$. The polynomial $Q_7^{'''}(x)$ has two simple real roots on $x\!\in\!\left(0,\nfrac{\pi}{2}\right)$ for $x= x_3$ and $ x= x_4$. Also holds that $Q_7^{'''}(0)>0$. That means that $Q_7^{'''}(x)>0$ for $ x\!\in\!(0, x_3)\cup(x_4,\infty)$ and $Q_7^{'''}(x)<0$ for $x\!\in\!(x_3, x_4)$ so the function $Q_7^{''}(x)$ is monotonically increasing for $ x\!\in\!(0, x_3)\cup(x_4,\infty)$ and monotonically decreasing for $ x\!\in\!(x_3, x_4)$. $Q_7^{''}(0)>0$, and $Q_7^{''}(0.98)>0$ and the function $Q_7^{''}(x)$ has no real roots on $ x\!\in\!\left(0,\nfrac{\pi}{2}\right)$. That means that $Q_7^{''}(x)>0$ for $x\!\in\!\left(0,\nfrac{\pi}{2}\right)$ so the function $Q_7^'(x)$ is monotonically increasing for $ x\!\in\!\left(0,\nfrac{\pi}{2}\right)$. $Q_7^'(0)<0$, $Q_7^'(0.98)>0$ the function $Q_7^'(x)$ has real root for $ x=0.30395 \ldots $ That means that $Q_7^'(x)<0$ for $x\!\in\!(0,0.30395 \ldots)$ and $Q_7^'(x)>0$ for $ x\!\in\!(0.30395 \ldots ,\infty)$ so the function $Q_7(x)$ is 