Câu hỏi 32

Ta có  \(\left[ f\left( x \right).f'\left( x \right) \right]'={{\left[ f'\left( x \right) \right]}^{2}}+f\left( x \right).f''\left( x \right)=15{{x}^{4}}+12x\)

Nguyên hàm 2 vế ta được \(f\left( x \right).f'\left( x \right)=3{{x}^{5}}+6{{x}^{2}}+C\)

Do \(f\left( 0 \right)=f'\left( 0 \right)=1\Rightarrow C=1\)

Tiếp tục nguyên hàm 2 vế ta được: \(\int{f\left( x \right)df\left( x \right)}=\int{\left( 3{{x}^{5}}+6{{x}^{2}}+1 \right)dx}\)

\(\Rightarrow \frac{{{f}^{2}}\left( x \right)}{2}=\frac{3{{x}^{6}}}{6}+\frac{6{{x}^{3}}}{3}+x+D=\frac{1}{2}{{x}^{6}}+2{{x}^{3}}+x+D\).

Do \(f\left( 0 \right)=1\Rightarrow D=\frac{1}{2}\Rightarrow {{f}^{2}}\left( x \right)=\frac{1}{2}{{x}^{6}}+2{{x}^{3}}+x+\frac{1}{2}\Rightarrow {{f}^{2}}\left( 1 \right)=4\)

Chọn A.