Question

Let f: R^2 to R be a function defined by

f(x,y,z) = { 1/(x-1) + 1/(y-1)+1/(z-1);

x≠1,y≠1,z≠1

{ 0 ; Otherwise

Calculate fxz(1,1,1)

Answer 1

Given f(x,y,z) = $\frac{1}{x-1} + \frac{1}{y-1} +\frac{1}{z-1}$  for $x\neq 1, y\neq 1, z\neq 1$ and otherwise f(x,y,z) = 0

f_x(x,y,z) = $\frac{\delta}{\delta x}(\frac{1}{x-1} + \frac{1}{y-1} +\frac{1}{z-1} )$

f_x(x,y,z) = $\frac{-1}{(x-1)^2}$

f_xz(x,y,z) = $\frac{\delta}{\delta z}$ f_x(x,y,z) = $\frac{\delta}{\delta z}$ $(\frac{-1}{(x-1)^2})$ = 0

therefore,

$\implies$ f_xz(x,y,z) = 0 for all (x,y,z) $\in$ R³

hence,

f_xz(1,1,1) = 0