Let f be a differentiable function R to R such that | f ,(x) , - … - Vidyadip

Question

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

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Answer

d
$|f(x)-f(y)| \leq 2|x-y|^{3 / 2}$

divide both side by $|x-y|$

$\left|\frac{f(x)-f(y)}{x-y}\right| \leq 2 .|x-y|^{1 / 2}$

Apply limit $x \rightarrow y$

$\left| {{f^\prime }(y)} \right| \le 0$

$ \Rightarrow {f^\prime }(y) = 0 \Rightarrow f(y) = c$

$ \Rightarrow f(x) = 1$

$\int_{0}^{1} 1 \cdot d x=1$

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