<s>[INST] Prove that $\frac{a^3}{b} + \frac{b^3}{a} \geqslant a^2 + b^2$ 
Prove that $$\frac{a^3}{b} + \frac{b^3}{a} \geqslant a^2 + b^2$$
for $a,b \in \mathbb{R}^+$. Establish when the equality holds.

My approach was to use AM-GM however trying both sides individually assert to $\geqslant 2ab.$ RHS is baby AM-GM if considered.
Edit:
From L.H.S -
$\dfrac{\frac{a^3}{b} + \frac{b^3}{a}}{2} \geqslant ab \implies \dfrac{a^3}{b} + \dfrac{b^3}{a} \geqslant 2ab$
 [/INST] Reduce to this:
$(a-b)(a^3-b^3)\ge 0$
Do you see why it is always satisfied?
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