TITLE: Mean value theorem application for multivariable functions
QUESTION [5 upvotes]: Define the function $f\colon \Bbb R^3\to \Bbb R$ by $$f(x,y,z)=xyz+x^2+y^2$$
  The Mean Value Theorem implies that there is a number $\theta$ with $0<\theta <1$ for which
  $$f(1,1,1)-f(0,0,0)=\frac{\partial f}{\partial x}(\theta, \theta, \theta)+\frac{\partial f}{\partial y}(\theta, \theta, \theta)+\frac{\partial f}{\partial z}(\theta, \theta, \theta)$$

This is the last question. I don't have any idea. Sorry for not writing any idea. How can we show MVT for this question?

REPLY [2 votes]: Following up on Peterson's hint, forget about the MVT for several variables and focus on the one dimensional version of it.
Consider the function $\varphi\colon [0,1]\to \Bbb R, t\to t^3+2t^2$.
The MVT guarantees the existence of $\theta\in ]0,1[$ such that $\varphi '(\theta)=\varphi(1)-\varphi (0)$.
Now try to relate $\varphi (1)$ with $f(1,1,1)$, $\varphi(0)$ with $f(0,0,0)$ and $\varphi '(\theta)$ with $\displaystyle \frac{\partial f}{\partial x}(\theta, \theta, \theta)+\frac{\partial f}{\partial y}(\theta, \theta, \theta)+\frac{\partial f}{\partial z}(\theta, \theta, \theta)$.