$x\ge -\dfrac{1}{2}; \dfrac{x}{y}>1$
Find the minimum of the function $\frac{2x^{3}+1}{4y\left ( x-y \right )}$
I thought of using a few values of $x$ and $y$, but that seemed extremely inefficient. I've seen many people describing AM-GM and Cauchy-Schwarz, but I'm not sure if that even applies to this problem. I have reinforced my understanding of AM-GM and Cauchy-Schwarz, but I'm still not sure how to solve this. Can someone guide me through this?
prove that $$\frac{2a^3+1}{4b(a-b)}\geq 3$$ and the equal sign holds if $$a=-\frac{1}{2},b=-\frac{1}{4}$$