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