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  • Solve! - Let f be a diffrential function from R to R such that, for all. If then is equal to : - Integral Calculus - JEE Main

Let f be a diffrential function from R to R such that |f(x)-f(y)|\leq 2|x-y|^{3/2}, for all x,y\epsilon R. If f(0)=1 then \int_{0}^{1}f^{2}(x)dx is equal to :

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Answers (1)

best_answer

 

lower and upper limit -

\int_{a}^{b}f\left ( x \right )dx= \left ( F\left ( x \right ) \right )_{a}^{b}

                = F\left ( b \right )-F\left ( a \right )

 

- wherein

Where a is lower and b is upper limit.

 

Given that 

|f(x) - f(y)| \leq 2|x-y|^{\frac{3}{2}}

Divide both sides by |x-y|

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

Apply limit x\rightarrow y

|f'(y)| \leq 0 \Rightarrow f'(y) = 0 \\ \Rightarrow f(y) = C \Rightarrow f(x) = 1 \\ \int_{0}^{1} 1\cdot dx = 1


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