Can you please help me how can solve the following nonlinear program by indicating some useful reference or videos relevant to this problem? With my limited knowledge in this field, I tried to solve it using the Karush-Kuhn-Tucker (KKT) conditions, but it is not working.
\begin{alignat*}{2} \text{maximize } \qquad & f(x_1, x_2 , x_3 , x_4, x_5, x_6)=\Big[8\left(x_1^{3} + x_2^{3} + x_3^{3} + x_4^{3} + x_5^{3} + x_6^{3}\right) \Big]^{\frac{2}{3}} \\ \text{subject to } \qquad & 4\left(x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}\right) \le 1\\ & 4\left(x_3^{2} + x_4^{2} + x_5^{2} + x_6^{2}\right) \le 1\\ & 4\left(x_1^{2} + x_2^{2} + x_5^{2} + x_6^{2}\right) \le 1\\ & x_1, x_2, x_3, x_4, x_5, x_6 \ge 0. \end{alignat*}
$$f=\left(8\sum_{k=1}^6x_k^2\right)^{\!\!\frac{2}{3}}\leq\left(4\cdot\frac{3}{4}\right)^{\!\!\frac{2}{3}}=\sqrt[3]{9}.$$ The equality occurs for $x_k=\frac{1}{4},$ which says that we got a maximal value.