KVPY Sample Paper KVPY Stream-SX Model Paper-11

Let f be a differentiable function R to R such that \[\left| f(x)-f(y) \right|\le 2{{\left| x-y \right|}^{\frac{3}{2}}},\]for all \[x,y\in R.\] If \[f(0)=1\]then \[\int\limits_{0}^{1}{{{f}^{2}}}(x)dx\]is equal to:

A) 0

B) \[\frac{1}{2}\]

C) 2

D) 1

Correct Answer: D

Solution :

\[\left| f(x)-f(y) \right|\le 2{{\left| x-y \right|}^{3/2}}\]

Dive both by \[\left| x-y \right|\]

\[\left| \frac{f(x)-f(y)}{x-y} \right|\le 2.{{\left| x-y \right|}^{1/2}}\]

Apply limit \[x\to y\]

\[\left| f'(y) \right|\le 0\]

\[\Rightarrow \] \[f'(y)=0\] \[\Rightarrow \] \[f(y)=c\] \[\Rightarrow \] \[f(x)=1\]

\[\int\limits_{0}^{1}{1.dx=1.}\]