Question

If $f$ is a real valued function such the $\left|f\left(x\right)-f\left(y\right)\right|\le {\left|x-y\right|}^3$ then $f^{\prime }\left(x\right)$

• A. $\left|\cos x\pi \right|$
• B. $\left|\sin \left[x\right]\pi \right|$
• C. $2\left|x\right|$
• D. not possible to find

Answer 1

The correct option is D not possible to find

$\left|f\right(x)-f(y\left)\right|\le {|x-y|}^3$

$\Rightarrow \frac{\left|f\right(x)-f(y\left)\right|}{|x-y|}\le {|x-y|}^2$

$\Rightarrow \underset{x\to y}{lim}\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|\le \underset{x\to y}{lim}{|x-y|}^2$

$\Rightarrow \underset{x\to y}{lim}\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|\le \underset{x\to y}{lim}{|x-y|}^2$

$\Rightarrow \left|f^{\prime }\right(x\left)\right|\le 0.$

which is not possible.

$\therefore f^{\prime }\left(x\right)$ is not possible to find.