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Question

ni=1f1(xi)=ni=1xi where f1 denotes the inverse of f.Find the AM of xis

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Solution

The condition,
ni=1f1(xi)=ni=1xi can be written as;
1nni=1f1(xi)=1nni=1f1xi
ie, AM of y-coordinates of f1=AM of x coordinates of f.
The given two conditions hold if and only if x2+3x3=x (point where f and f1 meet).
x2+2x3=0
So, x=3,+1
But x0x=1 (neglecting x=3)
Hence, we can write 1nni=1xi=1 which is the required result.

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