Unfortunately, Lagrange-type results do not hold for vector-valued functions. In other words, it is not true that there exists an \(\bar{\x}\) such that
\[\f(\x) = \f(\x_0) + \nabla \f(\bar{\x}) \Tr (\x - \x_0);\]such a point exists for each element of \(\f\) separately, but these points will not be the same.
However, big O versions of the Taylor series are still valid, as in the following theorem.
Theorem: Suppose \(\f:\real^d \to \real^k\) is twice differentiable on \(N_r(\x_0)\), and that \(\nabla^2 f\) is bounded on \(N_r(\x_0)\). Then for any \(\x \in N_r(\x_0)\),
\[\f(\x) = \f(\x_0) + \left[ \nabla \f(\x_0) + O(\norm{\x-\x_0})\one_{d \times k} \right] \Tr (\x - \x_0),\]where \(\one\) is a matrix of ones (i.e., every element equals one).
Proof: For any individual component \(j\) of \(\f\), there exists \(\bar{\x_j}\) on the line segment connecting \(\x\) and \(\x_0\) such that
\[\as{ f_j(\x) &= f_j(\x_0) + \nabla f_j(\x_0)\Tr(\x-\x_0) + \tfrac{1}{2}(\x-\x_0)\Tr \nabla^2 f_j(\bar{\x}_j)(\x-\x_0) \\ &= f_j(\x_0) + \left[\nabla f_j(\x_0) + \tfrac{1}{2}\nabla^2 f_j(\bar{\x}_j) (\x-\x_0) \right]\Tr (\x-\x_0) \\ &= f_j(\x_0) + \left[\nabla f_j(\x_0) + O(1) (\x-\x_0) \right]\Tr (\x-\x_0) \\ }\]because \(\nabla^2 f\) is bounded. Stacking these individual equations into a system of equations, we obtain the result stated in the theorem.
In statistics, this theorem is usually applied with respect to the score.