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Câu hỏi: Cho hàm số $y=f\left( x \right)$ liên tục, có đạo hàm trên $\left[ -1;0 \right]$. Biết $f'\left( x \right)=(3{{x}^{2}}+2x).{{e}^{-f\left( x \right)}}\forall x\in \left[ -1;0 \right]$. Tính giá trị biểu thức $A=f\left( 0 \right)-f\left( -1 \right)$.
A. $A=1.$
B. $A=0.$
C. $A=\dfrac{1}{e}.$
D. $A=-1.$
A. $A=1.$
B. $A=0.$
C. $A=\dfrac{1}{e}.$
D. $A=-1.$
$\begin{aligned}
& f'\left( x \right)=(3{{x}^{2}}+2x).{{e}^{-f\left( x \right)}}\Leftrightarrow \dfrac{f'\left( x \right)}{{{e}^{-f\left( x \right)}}}=3{{x}^{2}}+2x\Leftrightarrow f'\left( x \right).{{e}^{f\left( x \right)}}=3{{x}^{2}}+2x \\
& \Rightarrow \int\limits_{-1}^{0}{f'\left( x \right).{{e}^{f\left( x \right)}}}dx=\int\limits_{-1}^{0}{\left( 3{{x}^{2}}+2x \right)}dx \\
& \Leftrightarrow \left. {{e}^{f\left( x \right)}} \right|_{-1}^{0}=\left. \left( {{x}^{3}}+{{x}^{2}} \right) \right|_{-1}^{0}=0\Leftrightarrow {{e}^{f\left( 0 \right)}}-{{e}^{f\left( -1 \right)}}=0\Leftrightarrow {{e}^{f\left( 0 \right)}}={{e}^{f\left( -1 \right)}} \\
\end{aligned}$
Vì $y={{e}^{x}}$ là hàm số đồng biến ${{e}^{f\left( 0 \right)}}={{e}^{f\left( -1 \right)}}\Leftrightarrow f\left( 0 \right)=f\left( -1 \right)\Leftrightarrow A=f\left( 0 \right)-f\left( -1 \right)=0$
& f'\left( x \right)=(3{{x}^{2}}+2x).{{e}^{-f\left( x \right)}}\Leftrightarrow \dfrac{f'\left( x \right)}{{{e}^{-f\left( x \right)}}}=3{{x}^{2}}+2x\Leftrightarrow f'\left( x \right).{{e}^{f\left( x \right)}}=3{{x}^{2}}+2x \\
& \Rightarrow \int\limits_{-1}^{0}{f'\left( x \right).{{e}^{f\left( x \right)}}}dx=\int\limits_{-1}^{0}{\left( 3{{x}^{2}}+2x \right)}dx \\
& \Leftrightarrow \left. {{e}^{f\left( x \right)}} \right|_{-1}^{0}=\left. \left( {{x}^{3}}+{{x}^{2}} \right) \right|_{-1}^{0}=0\Leftrightarrow {{e}^{f\left( 0 \right)}}-{{e}^{f\left( -1 \right)}}=0\Leftrightarrow {{e}^{f\left( 0 \right)}}={{e}^{f\left( -1 \right)}} \\
\end{aligned}$
Vì $y={{e}^{x}}$ là hàm số đồng biến ${{e}^{f\left( 0 \right)}}={{e}^{f\left( -1 \right)}}\Leftrightarrow f\left( 0 \right)=f\left( -1 \right)\Leftrightarrow A=f\left( 0 \right)-f\left( -1 \right)=0$
Đáp án B.