Mean value theorem application for multivariable functions 
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? 

Hint: Consider $g(t):=f(t,t,t)$. What is $g'(t)$?
