kuntungw3

2022-01-24

Suppose that $x,y,z\in \mathbb{R}$ with $a>0$ a constant. Then, is the following true?
$f\left(x,y+cz,z\right)-f\left(x,y,z\right)=c{\int }_{0}^{1}\left({\partial }_{x}+z{\partial }_{y}\right)f\left(x,y+\theta z,z\right)d\theta .$
It looks like the mean value theorem for multivariable function but I don't know why ${\partial }_{x}$ appears.

marzembreax

Expert

That looks quite wrong. The right-hand side should be something like $cz{\int }_{0}^{1}{\partial }_{y}f\left(x,y+\theta cz,z\right)d\theta$

SaphypycleDapc5

Expert