Áp dụng BĐT Cauchy:
\(\sqrt{ab}=2\sqrt{\dfrac{a}{4}.b}\le\dfrac{a}{4}+b\)
\(\sqrt[3]{abc}=\sqrt[3]{\dfrac{a}{4}.b.4c}\le\dfrac{1}{3}\left(\dfrac{a}{4}+b+4c\right)=\dfrac{a}{12}+\dfrac{b}{3}+\dfrac{4c}{3}\)
\(\Rightarrow a+\sqrt{ab}+\sqrt[3]{abc}\le a+\dfrac{a}{4}+b+\dfrac{a}{12}+\dfrac{b}{3}+\dfrac{4c}{3}\)
\(\Leftrightarrow a+\sqrt{ab}+\sqrt[3]{abc}\le\dfrac{4}{3}\left(a+b+c\right)\)
\(\Leftrightarrow\dfrac{4}{3}\left(a+b+c\right)\ge\dfrac{4}{3}\)
\(\Rightarrow a+b+c\ge1\)
\(\Rightarrow M_{min}=1\) khi \(\left\{{}\begin{matrix}\dfrac{a}{4}=b=4c\\a+b+c=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=\dfrac{16}{21}\\b=\dfrac{4}{21}\\c=\dfrac{1}{21}\end{matrix}\right.\)
