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Answer the following question based on the information given below.
For three distinct real numbers $x, y$ and $z,$ let

  • $f(x, y, z) = \min(\max(x, y), \max(y, z), \max(z, x))$
  • $g(x, y, z) = \max(\min(x, y), \min(y, z), \min(z, x))$
  • $h(x, y, z) = \max(\max(x, y), \max(y, z), \max(z, x))$
  • $j(x, y, z) = \min(\min(x, y), \min(y, z), \min(z, x))$
  • $m(x, y, z) = \max(x, y, z)$
  • $n(x, y, z) = \min(x, y, z)$

Which of the following expressions is necessarily equal to $1?$

  1. $(f(x, y, z) – m(x, y, z))/(g(x, y, z) – h(x, y, z))$
  2. $(m(x, y, z) – f(x, y, z))/(g(x, y, z) – n(x, y, z))$
  3. $(j(x, y, z) – g(x, y, z))/h(x, y, z)$
  4. $(f(x, y, z) – h(x, y, z))/f(x, y, z)$
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Answer A

Say (x,y,z)=(1,3,4)

f(x,y,z)=3

g(x,y,z)=3

h(x,y,z)=4

j(x,y,z)=1

Now putting value we get the answer
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Need to show that other options are false also. Or else, prove f=g and m = h.
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