The original problem is to find nonnegative $\mathbf{w}=(w_1, w_2, w_3)$ that maximizes $$\pi_{\mathbf{w}}=\frac14(p_1-w_1)+\frac14(p_2-w_2)+\frac12(p_3-w_3)$$ subject to $$\frac14\sqrt{w_1}+\frac14\sqrt{w_2}+\frac12\sqrt{w_3}-600 \ge 400$$ and $$\frac14\sqrt{w_1}+\frac14\sqrt{w_2}+\frac12\sqrt{w_3}-600 \ge \frac12\sqrt{w_1}+\frac14\sqrt{w_2}+\frac14\sqrt{w_3}-0,$$ where $p_1$, $p_2$, and $p_3$ are given as $1$ million, $4$ million, and $9$ million dollars, respectively.
My professor solved this problem very easily. he said when $w_1=0$ and $w_3=5760000$, the second inequality is satisfied. So is the first inequality. Hence, $(w_1, w_2, w_3)=(0,0,5.76 \text{ million})$ and the total utility becomes $2.87$ million dollars.
First, I remove $p_i$'s in the objective function because it does not affect the optimal $w_i$'s.
Then, I normalize constant values.
Finally, I substitute $\sqrt{w_i}$'s with $x_i$'s in order to remove root terms.
Hence, I obtained a problem that finds nonnegative $\mathbf{x}=(x_1, x_2, x_3)$ that maximizes $$f(\mathbf{x})=x_1^2 + x_2^2+ 2x_3^2$$ subject to $$x_1+x_2+2x_3 \ge 4000$$ and $$-x_1 + x_3 \ge 2400.$$
I tried to solve the problem with five inequality constraints (additional $x_1\ge0$, $x_2\ge0$, and $x_3\ge0$) using KKT condition.
However, it is so complicated because there are three equality equations from $\nabla L$, where $L$ is a Laplacian equation, five original ineqaulities, five Lagrangian multiplier $\lambda_i$'s inequalities, and five complementary slackness equalities.
Is there any method to solve the problem easily?