Past exam of the options I course of the University of Oxford / 2009 / F / F1 / a / Solution

There are real Lagrange multipliers ${\lambda }_1,{\lambda }_2$ such thatEquivalently $Dg(x_0)={\lambda }^{T}Df(x_0)$. The rank assumption says the two constraint gradients are linearly independent; it rules out an abnormal constraint point.