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Let f: \mathbf{R} \rightarrow \mathbf{R} be a function defined by  f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}. If the function  g(x)=f(f(f(x)))+f(f(x)), then the greatest integer less than or equal to g(1) is______________.

Option: 1

2


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{ f(x) =\left[2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right]^{1 / 50}}
\mathrm{f(x) =\left[\left(2-x^{25}\right)\left(2+x^{25}\right)\right]^{1 / 50} }
         \mathrm{=\left(4-x^{50}\right)^ {1 / 50}}
\mathrm{f\left ( f\left ( x \right ) \right )= \left ( 4-\left ( \left ( 4-x^{50} \right ) ^{1/50}\right )^{50} \right )^{1/50}= x}
\mathrm{g(x) =f(f(f\left ( x \right )))+f(f(x))}
           \mathrm{=f(x)+x}

\mathrm{g(1) =f(1)+1=3^{1 / 50}+1 }
\mathrm{{[g(1)] } =\left[3^{1 / 50}+1\right]=2 }

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Rishi

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