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Given the problem to maximize $f(x), X=(x_1, x_2, \dots , x_n)$ subject to m number of in equality constraints. $g_i(x) \leq b_i$, i=1, 2, .... m including the non-negativity constrains $x \geq 0$. Which of the following conditions is a Kuhn-Tucker necessary condition for a local maxima at $\bar{x}$?

- $\frac{\partial L(\bar{X}, \bar{\lambda}, \bar{S})}{\partial x_j} = 0, j=1,2\dots m$
- $\bar{\lambda}_i [g_i(\bar{X})-b_i] = 0, i=1,2 \dots m$
- $g_i (\bar{X}) \leq b_i, i=1,2 \dots m$
- All of these

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