The total derivative and a line segment S

[Demon117]

Homework Statement

Let f be a differentiable function from an open set

$$U\subseteqR^{n}$$ into R. If

$$x,y\inU$$ and the segment $$S={(1-t)x+yt : t\in[0,1]}$$ is contained in U, show that

$$f(y)-f(x)=(Df)_{\xi}(y-x)$$ for some $$\xi\inS$$.

The Attempt at a Solution

The only direction I have is that I need to use the chain rule and the mean value theorem. Beyond that I do not know how to proceed.

 

[Demon117]

Homework Statement

Let f be a differentiable function from an open set

$$U\subseteqR^{n}$$ into R. If

$$x,y\inU$$ and the segment $$S={(1-t)x+yt : t\in[0,1]}$$ is contained in U, show that

$$f(y)-f(x)=(Df)_{\xi}(y-x)$$ for some $$\xi\inS$$.

The Attempt at a Solution

The only direction I have is that I need to use the chain rule and the mean value theorem. Beyond that I do not know how to proceed.

Totally screwed up the latex code here. Sorry.