Check whether the 1st-order necessary conditions for optimality hold at the optimum point (1, 0) of the following NLP:

$$min \; f(x) := -x_1$$ $$s.t. \;g_1(x) := -(1-x_1)^3 + x_2 \leq 0$$ $$g_2(x) := -x_2 \leq 0$$

In my attempted solution I used KKT conditions to find the optimum. For the g1, g2 = 0 case (which I believe is the only feasible one) I get x2=0 and x1=1 from the g2 and g1 solutions. However, I receive a contradiction from the stationarity condition for KKT multipliers as $3v_1(x_1-1)^2=1$ which is false.

What am I doing wrong here? I need to do this with a KKT approach but don't see my error

Check whether the 1st-order necessary conditions for optimality hold at the optimum point (1, 0) of the following NLP:

$$min \; f(x) := -x_1$$ $$s.t. \;g_1(x) := -(1-x_1)^3 + x_2 \leq 0$$ $$g_2(x) := -x_2 \leq 0$$

In my attempted solution I used KKT conditions to find the optimum. For the g1, g2 = 0 case (which I believe is the only feasible one) I get x2=0 and x1=1 from the g2 and g1 solutions. However, I receive a contradiction from the stationarity condition for KKT multipliers as $3v_1(x_1-1)^2=1$ which is false.

What am I doing wrong here? I need to do this with a KKT approach but don't see my error

Edit: As requested here is the KKT system I derived