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Câu hỏi: Giả sử hàm số $y=f\left( x \right)$ liên tục, đồng biến, nhận giá trị dương trên khoảng $\left( 0;+\infty \right)$ thỏa mãn $f\left( 3 \right)=\dfrac{4}{9}$ và ${{\left[ {f}'\left( x \right) \right]}^{2}}=\left( x+1 \right).f\left( x \right)$. Giá trị của $f\left( 8 \right)$ là
A. 49.
B. 36.
C. $5\sqrt{2}.$
D. $2\sqrt{10}.$
A. 49.
B. 36.
C. $5\sqrt{2}.$
D. $2\sqrt{10}.$
Ta có ${f}'\left( x \right)=\sqrt{\left( x+1 \right).f\left( x \right)}\Rightarrow \dfrac{{f}'\left( x \right)}{\sqrt{f\left( x \right)}}=\sqrt{x+1}$
$\Rightarrow \int\limits_{3}^{8}{\dfrac{{f}'\left( x \right)}{\sqrt{f\left( x \right)}}dx=\int\limits_{3}^{8}{\sqrt{x+1}dx}\Rightarrow \int\limits_{3}^{8}{\dfrac{1}{\sqrt{f\left( x \right)}}d\left[ f\left( x \right) \right]}=\int\limits_{3}^{8}{{{\left( x+1 \right)}^{\dfrac{1}{2}}}dx}}$
$\Rightarrow 2\sqrt{f\left( x \right)}\left| _{\begin{smallmatrix}
\\
3
\end{smallmatrix}}^{\begin{smallmatrix}
8 \\
\end{smallmatrix}} \right.=\dfrac{{{\left( x+1 \right)}^{\dfrac{1}{2}+1}}}{\dfrac{3}{2}}\Rightarrow 2\left[ \sqrt{f\left( 8 \right)}-\sqrt{f\left( 3 \right)} \right]=\dfrac{2}{3}\sqrt{\left( x+1 \right)}\left| _{\begin{smallmatrix}
\\
3
\end{smallmatrix}}^{\begin{smallmatrix}
8 \\
\end{smallmatrix}} \right.$
$\Rightarrow \sqrt{f\left( 8 \right)}-\sqrt{\dfrac{4}{9}}=\dfrac{19}{3}\Rightarrow f\left( 8 \right)=49.$
$\Rightarrow \int\limits_{3}^{8}{\dfrac{{f}'\left( x \right)}{\sqrt{f\left( x \right)}}dx=\int\limits_{3}^{8}{\sqrt{x+1}dx}\Rightarrow \int\limits_{3}^{8}{\dfrac{1}{\sqrt{f\left( x \right)}}d\left[ f\left( x \right) \right]}=\int\limits_{3}^{8}{{{\left( x+1 \right)}^{\dfrac{1}{2}}}dx}}$
$\Rightarrow 2\sqrt{f\left( x \right)}\left| _{\begin{smallmatrix}
\\
3
\end{smallmatrix}}^{\begin{smallmatrix}
8 \\
\end{smallmatrix}} \right.=\dfrac{{{\left( x+1 \right)}^{\dfrac{1}{2}+1}}}{\dfrac{3}{2}}\Rightarrow 2\left[ \sqrt{f\left( 8 \right)}-\sqrt{f\left( 3 \right)} \right]=\dfrac{2}{3}\sqrt{\left( x+1 \right)}\left| _{\begin{smallmatrix}
\\
3
\end{smallmatrix}}^{\begin{smallmatrix}
8 \\
\end{smallmatrix}} \right.$
$\Rightarrow \sqrt{f\left( 8 \right)}-\sqrt{\dfrac{4}{9}}=\dfrac{19}{3}\Rightarrow f\left( 8 \right)=49.$
Đáp án A.