x^3\\ [1.0 ex]
\!\!&\! \!&\!\! \!+\!\left(60\pi^4\!-\!2016\pi^2\right)x^2\!+\!\left(\!-\!12\pi^5\!+\!672\pi^3\right)x\!+\!\pi^6\!-\!84\pi^4\!+\!7560,
\end{array}$$ for $ x\!\in\!(0,c_1)$. The second derivate of the polynomial $\hat{P}_6(x)$ is the polynomial of $4^{th}$ degree $$\label{Ineq_39}
\begin{array}{rcl}
\hat{P}_6^{''}\!(x)\!\!&\!=\!&\!\!1920x^4\!-\!3840\pi x^3\!+\!\left(2880\pi^2\!-\!16128\right)x^2\!\\ [1.0 ex]
\!\!&\! \!&\!\! +\!\left(\!-960\pi^3\!+\!16128\pi \right)x+120\pi^4-4032\pi^2.
\end{array}$$ Factorization of $\hat {P}_6^{''}\!(x)$ is given by $$\label{Ineq_40}
\hat {P}_6^{''}\!(x)
=
24\left(20x^2\!-\!20\pi x\!+\!(5\pi^2\!-\!168)\right)\!\left(\pi\!-\!2x\right)^{2}
=
24\left(\pi\!-\!2x\right)^{2} \hat{P}_2(x),$$ where $$\label{Ineq_41}
\hat {P}_2(x)=20 x^2-20\pi x+(5\pi^2\!-168)$$ is quadratic polynomial with two simple real roots: $$\label{Ineq_42}
\hat {P}_2(x)=\alpha(x- x_1)(x- x_2),$$ with values $ \alpha=20$, $ x_1=-1.327\ldots $, $ x_2=4.469\ldots\;$. It holds that next inequalities are true $$\label{Ineq_43}
\begin{array}{rrcl}
\hat {P}_2(x)<0 & \mbox {for} & x\!\in\!(0,c_1)\subset(x_1, x_2)            \\[1.0 ex]
\Longrightarrow\; \hat{P}_6^{''}\!(x)<0 & \mbox {for} & x\!\in\!(0,c_1)\subset(x_1, x_2).
\end{array}$$ Therefore, for chosen interval $ x\!\in\!(0,c_1)$ the polynomial $\hat{P}_6^{{''}}(x)$ has no roots. Since $\hat{P}_6^{{''}}(0)<0$, the polynomial $\hat{P}_6^{{''}}(x)$ is negative function for $ x\!\in\!(0,c_1)$ and $\hat{P}_6^'(x)$ is monotonically decreasing function for $ x\!\in\!(0,c_1)$.

Furthermore, as the polynomial $\hat{P}_6^'(c_1)>0$ it follows that the polynomial $\hat{P}_6^'(x)$ is positive function for $ x\!\in\!(0,c_1)$, and the polynomial $\hat{P}_6(x)$ is monotonically increasing function for $ x\!\in\!(0,c_1)$. Because of $\hat{P}_6(0)>0$ we conclude following: $$\label{Ineq_44}
\begin{array}{rccl}
& \hat{P}_6(x)>0 & \mbox {for} & x\!\in\!(0,c_1) \\[1.0 ex]
\Longrightarrow & \hat{P}_7(x)>0 & \mbox {for} & x\!\in\!(0,c_1).
\end{array}$$ According to the Lemmas 2.3. and 2.4. and 