JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f\) be any function defined on \(R\) and let it satisfy the condition \(|f( x )-f( y )| \leq\left|( x - y )^{2}\right|, \forall( x , y ) \in R\) If \(f(0)=1,\) then
- A \(f(x)\) can take any value in \(R\)
- B \(f(x)< 0, \forall \,x \in R\)
- C \(f( x )=0, \forall \, x \in R\)
- D \(f( x )>0, \forall \, x \in R\)
Answer & Solution
Correct Answer
(D) \(f( x )>0, \forall \, x \in R\)
Step-by-step Solution
Detailed explanation
\(\left|\frac{f(x)-f(y)}{(x-y)}\right| \leq|(x-y)|\) \(x-y=h\) let \(\Rightarrow x=y+h\) \(\lim _{x \rightarrow 0}\left|\frac{f(y+h)-f(y)}{h}\right| \leq 0\) \(\Rightarrow\left|f^{\prime}( y )\right| \leq 0 \Rightarrow f^{\prime}( y )=0\) \(\Rightarrow f( y )= k (\) constant…
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