Proof using f(x1,y1)-f(x0,y0)=fx(x*,y*)(x1-x0)+fy(x*,y*)(y1-y0)

[brandy]

1. Homework Statement [/prove f(x1,y1)-f(x0,y0)=fx(x*,y*)(x1-x0)+fy(x*,y*)(y1-y0)
prove there exists a point (x*,y*)
if fx and fy are all continuous on a circular region and contain A(x0,y0) and B(x1,y1)

Homework Equations

The Attempt at a Solution

I'm thinking mean value theorem but honestly i have no idea how to do this.
i don't even know how to properly apply the mean value theorem to this since there are two variables changing and the teacher said something about parametrization.

 

[Ray Vickson]

1. Homework Statement [/prove f(x1,y1)-f(x0,y0)=fx(x*,y*)(x1-x0)+fy(x*,y*)(y1-y0)
prove there exists a point (x*,y*)
if fx and fy are all continuous on a circular region and contain A(x0,y0) and B(x1,y1)

Homework Equations

The Attempt at a Solution

I'm thinking mean value theorem but honestly i have no idea how to do this.
i don't even know how to properly apply the mean value theorem to this since there are two variables changing and the teacher said something about parametrization.

Look at the univariate function g(t) = f(x_0 + t(x_1 - x_0), y_0 + t(y_1 - y_0)) on 0 \leq t \leq 1.

RGV