f, h cont diff. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at the point (0,0).

Question

Suppose that the functions $f: \mathbb R^2 \to \mathbb R$ and $h: \mathbb R^2 \to \mathbb R$ are continuously differentiable. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at the point (0,0).

Answer 1

See https://en.wikipedia.org/wiki/Order_of_approximation#First-order.

First-order approximation means coincidence of functions and first derivatives in the point, so $$f(0,0) = h(0,0)$$ $$\cdots = Df(0,0) = Dh(0,0) = \cdots$$ where you can write $Df(0,0)$, $Dh(0,0)$ using partial derivatives...