Question

Cho a+b=2

CMR : \(\sqrt[3]{a}+\sqrt[3]{b}\le2\)

Answer 1

Áp dụng bất đẳng thức Holder, ta có: 

\(\left[\left(\sqrt[3]{a}\right)^3+\left(\sqrt[3]{b}\right)^3+1^3\right].\left(1^3+1^3+1^3\right).\left(1^3+1^3+1^3\right)\ge\left(\sqrt[3]{a}.1.1+\sqrt[3]{b}.1.1+1.1.1\right)^3\)

<=>\(\left(a+b+1\right).9\ge\left(\sqrt[3]{a}+\sqrt[3]{b}+1\right)^3\)

Vì a+b=3

=>\(\left(\sqrt[3]{a}+\sqrt[3]{b}+1\right)^3\le27\)

<=>\(\sqrt[3]{a}+\sqrt[3]{b}+1\le3\)

<=>\(\sqrt[3]{a}+\sqrt[3]{b}\le2\)

Dấu "=" xảy ra khi: a=b=1

=>ĐPCM

Answer 2

nhầm a+b=2 đó nha