If x, y, z are different and
$increment space equals space open vertical bar table row straight x cell space space space space straight x squared end cell cell space 1 plus straight x cubed end cell row straight y cell space space space straight y squared end cell cell space 1 plus straight y cubed end cell row straight z cell space space straight z squared end cell cell space 1 plus straight z cubed end cell end table close vertical bar space equals space 0 comma space space then space show space that space 1 plus xyz space equals 0$

Solution

increment space equals space open vertical bar table row straight x cell space space space straight x squared end cell cell space space 1 plus straight x cubed end cell row straight y cell space space space straight y squared end cell cell space space 1 plus straight y cubed end cell row straight z cell space space straight z squared end cell cell space space 1 plus straight z cubed end cell end table close vertical bar space equals space open vertical bar table row straight x cell space space space straight x squared end cell cell space 1 end cell row straight y cell space space straight y squared end cell cell space 1 end cell row straight z cell space space straight z squared end cell cell space 1 end cell end table close vertical bar plus space open vertical bar table row straight x cell space space space straight x squared end cell cell space space space straight x cubed end cell row straight y cell space space straight y squared end cell cell space space straight y cubed end cell row straight z cell space space straight z squared end cell cell space space straight z cubed end cell end table close vertical bar space space space equals space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space space straight z end cell cell space space straight z squared end cell end table close vertical bar plus space straight x space straight y space straight z space space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 cell space straight y end cell cell space space straight y squared end cell row 1 cell space straight z end cell cell space space straight z squared end cell end table close vertical bar space equals space left parenthesis 1 plus xyz right parenthesis space open vertical bar table row 1 cell space space space straight x end cell cell space space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space space straight z end cell cell space space space straight z squared end cell end table close vertical bar space equals space left parenthesis 1 plus xyz right parenthesis thin space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis Now space increment space equals space 0 rightwards double arrow space space space space left parenthesis 1 plus xyz right parenthesis thin space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis rightwards double arrow space space space space space space space 1 plus xyz space equals 0 space space space space space space space space space space space space space open square brackets table row cell because space space space straight x comma space straight y comma space straight z space are space all space different end cell row cell because space straight x space minus space straight y space not equal to 0 comma space space straight y minus straight z not equal to 0 comma space space straight z minus straight x not equal to 0 end cell end table close square brackets

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