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Question

Let f:RR be a continuous decreasing function. A point x0R is said to be a fixed point of f if f(x0)=x0.
The number of distinct 3-tuples (x,y,z) satisfying the system
x=f(y),y=f(z),z=f(x)
equals

A
0
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B
1
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C
2
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D
infinitely many
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Solution

The correct option is B 1
x=f(y)y=f(z)z=f(x)

x=f(f(z))
=f(f(f(x)))
=(fff)(x) ...(1)
Similarly,
y=(fff)(y) ...(2)
z=(fff)(z) ...(3)

Now, let x,yDf=R and x<y
f(x)f(y)
f(f(x))f(f(y))
​​​​​​​f(f(f(x)))f(f(f(y)))
​​​​​​​​​​​​​​​​​​​​​(fff)(x)(fff)(y)
​​​​​​​​​​​​​​​​​​​​​fff is decreasing function on R.
​​​Therefore, it has only one fixed point.

Since, fff has only one fixed point.
x=y=z

Hence, there is only one distinct 3-tuples (x,y,z) that satisfying the system of equations.

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