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?

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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 $\displaystyle \frac{\partial f}{\partial x}(\theta, \theta, \theta)+\frac{\partial f}{\partial y}(\theta, \theta, \theta)+\frac{\partial f}{\partial z}(\theta, \theta, \theta)$.

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