<s>[INST] Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $ If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I proceed to the results? Please help me to proceed. Thanks in advance.
 [/INST] We need to prove that $$\sum_{cyc}\left(a^3-a^{\frac{7}{3}}b^{\frac{1}{3}}c^{\frac{1}{3}}\right)\geq0$$ and since $$(3,0,0)\succ\left(\frac{7}{3},\frac{1}{3},\frac{1}{3}\right),$$ it's true by Muirhead.
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