Cho hàm số $f\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx,\left(...

Ta có: $f\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx\Rightarrow {f}'\left( x \right)=5a{{x}^{4}}+3b{{x}^{2}}+c$
Nên: $g\left( x \right)=f\left( 1-2x \right)+2.f\left( x+4 \right)+m\Rightarrow {g}'\left( x \right)=-2.{f}'\left( 1-2x \right)+2.{f}'\left( x+4 \right)$
$\begin{aligned}
& \Rightarrow {g}'\left( x \right)=-2\left[ 5a{{\left( 1-2x \right)}^{4}}+3b{{\left( 1-2x \right)}^{2}}-5a{{\left( x+4 \right)}^{4}}-3b{{\left( x+4 \right)}^{2}} \right] \\
& \text{ }=-2\left[ 15a\left( x-5 \right)\left( x+1 \right)\left( 5{{x}^{2}}+4x+17 \right)+9b\left( x-5 \right)\left( x+1 \right) \right] \\
& \text{ =}-2\left( x-5 \right)\left( x+1 \right)\left[ 15a\left( 5{{x}^{2}}+4x+17 \right)+9b \right] \\
\end{aligned}$
Với $a>0,b>0\Rightarrow {g}'\left( x \right)\ge 0\forall x\in \left[ -1;5 \right]\Rightarrow g\left( x \right)$ đồng biến trên $\left[ -1;5 \right]$
Vì $f\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx\Rightarrow f\left( x \right)$ là hàm số lẻ nên $f\left( -9 \right)=-f\left( 9 \right)$
$\left\{ \begin{aligned}
& g\left( 5 \right)=f\left( -9 \right)+2f\left( 9 \right)+m=f\left( 9 \right)+m=m+81 \\
& g\left( -1 \right)=3f\left( 3 \right)+m=m-7 \\
\end{aligned} \right.$
TH1: $\left( m-7 \right)\left( m+81 \right)>0\Leftrightarrow m<-81\vee m>7$
Khi đó: $\underset{\left[ -1;5 \right]}{\mathop{\max }} \left| g\left( x \right) \right|+\underset{\left[ -1;5 \right]}{\mathop{\min }} \left| g\left( x \right) \right|=\left| m-7 \right|+\left| m-81 \right|\ge \left| \left( 7-m \right)+m-81 \right|=88>86$
Không thỏa mãn điều kiện bài toán.
TH2: $\left( m-7 \right)\left( m+81 \right)\le 0\Leftrightarrow -81\le m\le 7.$
$\underset{\left[ -1;5 \right]}{\mathop{\min }} \left| g\left( x \right) \right|=0\Rightarrow \underset{\left[ -1;5 \right]}{\mathop{\max }} \left| g\left( x \right) \right|=86$
$\left\{ \begin{aligned}
& \left| m+81 \right|=86 \\
& \left| m+81 \right|\ge \left| m-7 \right| \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \left[ \begin{aligned}
& m=5 \\
& m=-167 \\
\end{aligned} \right. \\
& \left| m+81 \right|\ge \left| m-7 \right| \\
\end{aligned} \right.\Leftrightarrow m=5.$
$\left\{ \begin{aligned}
& \left| m-7 \right|=86 \\
& \left| m-7 \right|\ge \left| m+81 \right| \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \left[ \begin{aligned}
& m=-79 \\
& m=93 \\
\end{aligned} \right. \\
& \left| m-7 \right|\ge \left| m+81 \right| \\
\end{aligned} \right.\Leftrightarrow m=-79.$
Vậy tổng các giá trị $m$ thỏa mãn là: $S=-79+5=-74.$