Prove Derivative Matrices Mapping F:Rn→Rm is Ax+c

[Appa]

Homework Statement

Suppose that the mapping F:Rⁿ\rightarrowR^m is continuously differentiable and that there is a fixed mxn matrix A so that
DF(x)=A for every x in Rⁿ.

Prove that then F is a mapping such that F(x)=Ax+c for some c\inR^m

Homework Equations

DF(x)_ij= \partialF_i(x)/\partialx_j (the ijth entry of the derivative matrix)

The Attempt at a Solution

I tried to solve this with the first-order approximation F(x)=F(x_o)+F´(x_o)(x-x_o) +\epsilon(x-x_o)||x-x_o||

Where \epsilon(x-x_o)||x-x_o|| approaches naught.
But it didn't amount to anything sensible and I'm not sure what to do... What should I try next?

 

[CompuChip]

I tried to solve this with the first-order approximation F(x)=F(x_o)+F´(x_o)(x-x_o) +\epsilon(x-x_o)||x-x_o||

Actually, when you write F´(x_o)(x-x_o) like in a Taylor series, you mean
DF(x_o) . (x-x_o)
which is a multiplication of a matrix by a vector. You know that DF(x_o) = A... if your expansion is otherwise correct that should enable you to actually identify c explicitly.