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Statement 1: Let f: RvecR be a real-valu...

Statement 1: Let `f: RvecR` be a real-valued function `AAx ,y in R` such that `|f(x)-f(y)|<=|x-y|^3` . Then `f(x)` is a constant function. Statement 2: If the derivative of the function w.r.t. `x` is zero, then function is constant.

Text Solution

Verified by Experts

`"We have "|f(x)-f(y)|le|x-y|^(3),xney`
`therefore" "|(f(x)-f(y))/(x-y)|le|x-y|^(2)`
`rArr" "underset(yrarrx)lim|(f(x)-f(y))/(x-y)|leunderset(yrarrx)lim|x-y|^(2)`
`rArr" "|underset(yrarrx)lim(f(x)-f(y))/(x-y)|le|underset(yrarrx)lim(x-y)^(2)|`
`rArr" "|f'(x)|le0`
`rArr" "|f'(x)|=0" "(because|f'(x)|ge0)`
`therefore" "f'(x)=0`
`rArr" "f(x)=c" (constant)"`
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