How to solve $6x^{2/3}-12-2x=0$? How to solve  $6x^{2/3}-12-2x=0$? I need it to prove that the graphic of $y=6x^{2/3}$ never meets the graphic of function $y=12+2x$.
We have $x+6 = 3x^{2/3}$ i.e $(x+6)^3 = 27x^2$. This has a real solution. You won't be able to prove what you wish.
Let $x^{\frac{1}{3}}=y$.

Thus, $x&gt;0$, $y&gt;0$ and we need to solve
$$y^3-3y^2+6=0.$$ 
But, by AM-GM
$$y^3+6=2\left(\frac{y^3}{2}\right)+6\geq3\sqrt[3]{6\left(\frac{y^3}{2}\right)^2}&gt;3y^2,$$ which says that our equation has no real roots.