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Let f be any function defined on R and let it satisfy the condition : \left | f\left ( x \right )-f\left ( y \right ) \right |\leq \left | \left ( x-y \right )^{2} \right |,\forall \left ( x,y \right )\; \epsilon \; R If f(0)=1, then :
 
Option: 1 f(x) can take any value in R
Option: 2 f(x) > 0, \forall\; x\; \epsilon \; R
Option: 3 f(x) < 0, \forall\; x\; \epsilon \; R  
Option: 4 f(x) = 0, \forall\; x\; \epsilon \; R

Answers (1)

best_answer

Given that

\\|f(x)-f(y)| \leq\left|(x-y)^{2}\right|\\\left|\frac{f(x)-f(y)}{(x-y)}\right| \leq|(x-y)|\\\text{Let }x -y=h\Rightarrow x=y+h\\\lim _{x \rightarrow 0}\left|\frac{f(y+h)-f(y)}{h}\right| \leq 0\\

\\\Rightarrow\left|f^{\prime}(y)\right| \leq 0 \Rightarrow f^{\prime}(y)=0 \\ \Rightarrow f(y)=k(\text { constant }) \\ \text { and } f(0)=1 \text { given } \\ \text { So, } f(y)=1 \Rightarrow f(x)=1

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himanshu.meshram

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