\(f^2\left(x\right).f'\left(x\right)=x.e^x\)
Lấy nguyên hàm 2 vế:
\(\int f^2\left(x\right).f'\left(x\right)dx=\int x.e^xdx\)
\(\Rightarrow\dfrac{1}{3}f^3\left(x\right)=\left(x-1\right)e^x+C\)
Thay \(x=1\)
\(\Rightarrow\dfrac{f^3\left(1\right)}{3}=C\Rightarrow C=\dfrac{1}{3}\)
\(\Rightarrow\dfrac{1}{3}f^3\left(x\right)=\left(x-1\right)e^x+\dfrac{1}{3}\Rightarrow f^3\left(x\right)=3\left(x-1\right)e^x+1\)
\(f\left(x\right)+1=0\Leftrightarrow f\left(x\right)=-1\Leftrightarrow f^3\left(x\right)=-1\)
\(\Leftrightarrow3\left(x-1\right)e^x+1=-1\Rightarrow3\left(x-1\right)e^x+2=0\)
Xét hàm \(g\left(x\right)=3\left(x-1\right)e^x+2\Rightarrow g'\left(x\right)=3x.e^x=0\Rightarrow g'\left(x\right)=0\) có đúng 1 nghiệm
\(\Rightarrow g\left(x\right)=0\) có tối đa 2 nghiệm
\(g\left(0\right)=-1< 0\) ; \(g\left(1\right)=2>0\) ; \(g\left(-2\right)=-\dfrac{9}{e^2}+2>0\)
\(\Rightarrow\left\{{}\begin{matrix}g\left(0\right).g\left(1\right)< 0\\g\left(0\right).g\left(-2\right)< 0\end{matrix}\right.\) \(\Rightarrow g\left(x\right)=0\) có đúng 2 nghiệm
