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Câu hỏi: Tìm giá trị lớn nhất (max) và giá trị nhỏ nhất (min) của hàm số $y=x+\dfrac{1}{x}$ trên đoạn $\left[ \dfrac{3}{2}; 3 \right]$.
A. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{13}{6}$.
B. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=2$.
C. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{16}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=2$.
D. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{5}{2}$.
A. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{13}{6}$.
B. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=2$.
C. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{16}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=2$.
D. $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{5}{2}$.
Ta có:
${y}'=1-\dfrac{1}{{{x}^{2}}}$, ${y}'=0$ $\Leftrightarrow \left[ \begin{aligned}
& x=-1\notin \left[ \dfrac{3}{2}; 3 \right] \\
& x=1\notin \left[ \dfrac{3}{2}; 3 \right] \\
\end{aligned} \right.$.
$y\left( \dfrac{3}{2} \right)=\dfrac{13}{6}$, $y\left( 3 \right)=\dfrac{10}{3}$.
Suy ra $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{13}{6}$.
${y}'=1-\dfrac{1}{{{x}^{2}}}$, ${y}'=0$ $\Leftrightarrow \left[ \begin{aligned}
& x=-1\notin \left[ \dfrac{3}{2}; 3 \right] \\
& x=1\notin \left[ \dfrac{3}{2}; 3 \right] \\
\end{aligned} \right.$.
$y\left( \dfrac{3}{2} \right)=\dfrac{13}{6}$, $y\left( 3 \right)=\dfrac{10}{3}$.
Suy ra $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\max }} y=\dfrac{10}{3}$, $\underset{\left[ \dfrac{3}{2}; 3 \right]}{\mathop{\min }} y=\dfrac{13}{6}$.
Đáp án A.