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Help: Functions

If \(f:\mathbb R\longrightarrow \mathbb R\) satisfying \(\left| f\left( x \right) -f\left( y \right) \right| \le { \left| x-y \right| }^{ 3 }\), for all \(x,y\in \mathbb R\) and \(f\left( 2 \right) =5\), then find \(f\left( 4 \right)\).

Note by Akhilesh Prasad
1 year, 9 months ago

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I replaced x by x+h and y by x and took \(\lim_{x\rightarrow0}\) i.e \[|\lim_{x\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}|\leq0\] and since Modulus of a real no. cannot be less than 0, we get
\[f'(x)=0 \Rightarrow f(x)=Constant=C(let)\] And since f(2)=5 \(\Rightarrow \) f(x)=5..
Or f(4)=5.
Maybe I have done some mistake as I am lacking time at the moment.. Anyways can you find the error??

Rishabh Cool - 1 year, 9 months ago

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Pretty much did the same thing, seems like they misprinted the answer. Thanks for solving.

Akhilesh Prasad - 1 year, 9 months ago

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Why must the function be differentiable?

Calvin Lin Staff - 1 year, 9 months ago

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I am getting f(4)= 5. Is it correct?

Rishabh Cool - 1 year, 9 months ago

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Thanks a lot for solving all my doubts patiently.

Akhilesh Prasad - 1 year, 9 months ago

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That's what i was getting but the answer given is 25. If you could please write your solution so i can check whether my approach is correct.

Akhilesh Prasad - 1 year, 9 months ago

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@Rishabh Cool,@parv mor. Please if you could post a solution to the question.

Akhilesh Prasad - 1 year, 9 months ago

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