I've been working on a minimization problem for a while, involving "simple" conditions, but haven't been able to figure it out.

I've tried using Lagrange Multipliers and KKT, but the presence of $\frac{1}{\gamma^2}$ and $\frac{1}{n^2}$ in $\nabla f$ seems to make it impossible.

I need to minimize $f(n, \gamma) = \frac{3}{n} + \frac{2}{\gamma}$ with the primary constraint $g(n, \gamma) = \gamma \cdot(n^D + 3n^{D-1}) + 2n^D \le M$. Additionally, $\gamma, n > 0$.

Where $M, D$ are constants, and $\gamma, n$ are real variables to optimize over.

Should I be looking for a different optimization method?

So far, I have

$$\nabla f = \bigl< -\frac{3}{n^2}, -\frac{2}{\gamma^2} \bigr>$$

$$\nabla g(n, \gamma) = \bigl< \gamma\cdot(Dn^{D-1} + 3(D-1)n^{D-2}) + 2Dn^{D-1}, n^D + 3n^{D-1} \bigr>$$.