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Câu hỏi: Cho $a,b>0$ thỏa mãn: ${{a}^{\dfrac{1}{2}}}>{{a}^{\dfrac{1}{3}}},{{b}^{\dfrac{2}{3}}}>{{b}^{\dfrac{3}{4}}}$ khi đó khẳng định nào sau đây là đúng?
A. $0<a<1,b>1$.
B. $0<b<1<a$.
C. $0<a<1,0<b<1$.
D. $a>1,b>1$.
A. $0<a<1,b>1$.
B. $0<b<1<a$.
C. $0<a<1,0<b<1$.
D. $a>1,b>1$.
Ta có $\left\{ \begin{aligned}
& {{a}^{\dfrac{1}{2}}}>{{a}^{\dfrac{1}{3}}}\Leftrightarrow a>1\left( \text{do }\dfrac{1}{2}>\dfrac{1}{3} \right) \\
& {{b}^{\dfrac{2}{3}}}>{{b}^{\dfrac{3}{4}}}\Leftrightarrow 0<b<1\left( \text{do }\dfrac{2}{3}<\dfrac{3}{4} \right) \\
\end{aligned} \right.\Leftrightarrow 0<b<1<a$.
& {{a}^{\dfrac{1}{2}}}>{{a}^{\dfrac{1}{3}}}\Leftrightarrow a>1\left( \text{do }\dfrac{1}{2}>\dfrac{1}{3} \right) \\
& {{b}^{\dfrac{2}{3}}}>{{b}^{\dfrac{3}{4}}}\Leftrightarrow 0<b<1\left( \text{do }\dfrac{2}{3}<\dfrac{3}{4} \right) \\
\end{aligned} \right.\Leftrightarrow 0<b<1<a$.
Đáp án B.