Question

If $f:R\to R$ satisfies $|f\left(x\right)-f\left(y\right)|\le {|x-y|}^3$ and $f\left(4\right)=192$ then $f\left(7\right)$ is equal to

Answer 1

$|f\left(x\right)-f\left(y\right)|\le {|x-y|}^3\Rightarrow \frac{|f\left(x\right)-f\left(y\right)|}{|x-y|}\le {|x-y|}^2\underset{x\to y}{lim}\frac{|f\left(x\right)-f\left(y\right)|}{|x-y|}\le \underset{x\to y}{lim}{|x-y|}^2{f}^{\prime }\left(x\right)\le 0f^{\prime }\left(x\right)=0thatmeansi{t}^{\prime }saconstantfunctionsof\left(4\right)=f\left(7\right)f\left(7\right)=192$