1f(x) can be proved only to be constant function only if its ddxf(x) = 0 here
the mod helps one to do so
Let f:R→R such that for all x,y belongs R
where n belongs N.Prove that f(x) is a constant function.
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6 Answers
1
11Ah....this is nice.....
See from L.M.V.t....we can say there should be some 'k' in the interval (x,y) whre the slope of tangent should equal the slope of the line joining the end points.....
So f'(k)=|f(y)-f(x)|y-x
Since Numerator always ≤D, f'(k)≤1....
Choose y=x+Δx.....Now making tis interval arbitarily small, we can say f'(k)→0......Get it/
Well u may raise an objection whether f(x) is differentiable or not, that can be settled if u use the condition given in the qsn and take same intervals......only thing u shall have now is...
|f(x)-f(x+Δx)|≤|Δx|7
Divide both sides by Δx, and use the fact that Δx→0....that gives f'(x)=0 identically.
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