# 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?

#### Solutions Collecting From Web of "Mean value theorem application for multivariable functions"

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 $g'(t)$?