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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 $$f\left( 4 \right)$$.

1 year, 1 month 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?? · 1 year, 1 month ago

Pretty much did the same thing, seems like they misprinted the answer. Thanks for solving. · 1 year, 1 month ago

Why must the function be differentiable? Staff · 1 year, 1 month ago

I am getting f(4)= 5. Is it correct? · 1 year, 1 month ago

Thanks a lot for solving all my doubts patiently. · 1 year, 1 month ago

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. · 1 year, 1 month ago