Tống tất cả các giá trị nguyên của tham số \(m\) để hàm số \(y=3{{x}^{4}}+8m{{x}^{3}}+12\left( {{m}^{2}}-2m \right){{x}^{2}}\) có ba cực trị là:
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Phương pháp giải
Hàm số đa thức bậc 4 có 3 điểm cực trị khi phương trình đạo hàm có 3 nghiệm phân biệt.
Tìm điều kiện của tham số để hàm số bậc bốn trùng phương có điểm cực trị.
Lời giải
Ta có: \(y=3{{x}^{4}}+8m{{x}^{3}}+12\left( {{m}^{2}}-2m \right){{x}^{2}}\)
\({{y}^{\prime }}=12{{x}^{3}}+24m{{x}^{2}}+24\left( {{m}^{2}}-2m \right)=12x.\left[ {{x}^{2}}+2mx+2\left( {{m}^{2}}-2m \right) \right]\).
\({{y}^{\prime }}=0\Leftrightarrow \left[ \begin{array}{*{35}{l}} x=0 \\ {{x}^{2}}+2mx+2\left( {{m}^{2}}-2m \right)=0(*) \\\end{array} \right.\)
Để hàm số có 3 điểm cực trị thì phương trình \({{y}^{\prime }}=0\) có 3 nghiệm phân biệt.
Do đó, phương trình (*) phải có 2 nghiệm phân biệt khác 0.
\(\begin{align} & \Leftrightarrow \left\{ \begin{array}{*{35}{l}} \Delta >0 \\ {{0}^{2}}+2m\cdot 0+2({{m}^{2}}-2m)\ne 0 \\\end{array} \right. \\ & \Leftrightarrow \left\{ \begin{array}{*{35}{l}} 4{{m}^{2}}-8\left( {{m}^{2}}-2m \right)>0 \\ m\ne 0;m\ne 2 \\\end{array} \right. \\ & \Leftrightarrow \left\{ \begin{array}{*{35}{l}} 0<m<4 \\ m\ne 0;m\ne 2 \\\end{array} \right. \\ & \Leftrightarrow \left[ \begin{array}{*{35}{l}} m=1 \\ m=3 \\\end{array} \right. \\ \end{align}\)
Vậy tổng các giá trị của \(m\) thỏa mãn là: 4.
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