Let ' f ' is a differentiable function such that f ′ ( x ) = f ( x ) + ∫ 0 2 f ( x ) d x , f ( 0 ) = 3 4 − e 2 then the value of [ f ( 2 )] where [.] denotes the greatest integer ≤ x is.

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Let ' f ' is a differentiable function such that f ′ ( x ) = f ( x ) + ∫ 0 2 ​ f ( x ) d x , f ( 0 ) = 3 4 − e 2 ​ then the value of [ f ( 2 )] where [.] denotes the greatest integer ≤ x is.

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\eqalign{ & Given\,f&#x27;(x) = f(x) + A\,where\,A = \int\limits_0^2 {f(x)dx} \cr & Solving - (1) \cr & f(x) = \lambda ({e^2} - 1) + {{4 - {e^2}} \over 3} \cr & \int\limits_0^2 {f(x)} \,dx = A \Rightarrow \lambda = 1\,\,and\,\,A = {{{e^2} - 1} \over 3} \cr & f(x) = {e^x} - 1 + {{4 - {e^2}} \over 3} = {e^x} - {1 \over 3}\left( {{e^2} - 1} \right) \cr & f(2) = {{2{e^2} + 1} \over 3}\,\,\,\,[f(2)] = 5 \cr}

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JEE Advanced (2028)