I want to maximize and minimize $$h(a,b) = a + b$$ given the constraint $$g(a,b) = a^{\frac{1}{3}}b^{\frac{2}{3}} = l$$
I'm trying to use Lagrange multipliers. Here's what I did:
\begin{align} \frac{\partial g}{\partial a} = \lambda \frac{\partial h}{\partial a} \implies 1 = \lambda(\frac{1}{3}\frac{1}{a^{\frac{2}{3}}}b^{\frac{2}{3}})\\\\ \frac{\partial g}{\partial b} = \lambda \frac{\partial h}{\partial b} \implies 1 = \lambda(\frac{2}{3}a^{\frac{1}{3}}\frac{1}{b^{\frac{1}{3}}})\\ \end{align}
Then I solved the second equation for $b$:
\begin{align} (b^{\frac{1}{3}})^3 &= (\frac{2}{3} \lambda a^{\frac{1}{3}})^3\\ b &= \frac{8}{27} \lambda^3 a \end{align}
Plugged this value for $b$ into the constraint equation and solved for $a$:
\begin{align} a^{\frac{1}{3}}b^{\frac{2}{3}} &= l\\ a^{\frac{1}{3}}(\frac{8}{27} \lambda^3 a)^{\frac{2}{3}} &= l\\ a^{\frac{1}{3}}(\frac{2}{3} \lambda a^{\frac{1}{3}})^2 &= l\\ a^{\frac{1}{3}}(\frac{4}{9} \lambda^2 a^{\frac{2}{3}}) &= l\\ \frac{4}{9} \lambda^2 a &= l\\ a &= \frac{9}{4} \frac{l}{\lambda^2} \end{align}
Then I took this value for $a$ and plugged it into the expression I obtained for $b$ to get:
\begin{align} b &= \frac{8}{27} \lambda^3 a\\ b &= \frac{8}{27} \lambda^3 (\frac{9}{4}\frac{l}{\lambda^2})\\ b &= \frac{2}{3} \lambda l \end{align}
So now I have:
\begin{align} a &= \frac{9}{4} \frac{l}{\lambda^2} \\ b &= \frac{2}{3} \lambda l \\ a^{\frac{1}{3}}b^{\frac{2}{3}} &= l \end{align}
But now I'm stuck. And I'm not sure I am correct so far.