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Let f: R R be a function satisfying the relation 4 f(3-x)+3 f(x)=x^2 for any real x . Find the value of f(27)-f(25) to the nearest integer. (Here R denotes the set of real numbers.)

Correct answer

8

Step-by-step solution

aligned & 4 f(3-x)+3 f(x)=x^2 x R & 4 f(3-(3-x))+3 f(3-x)=(3-x)^2 & =4 f(x)+3 f(3-x)=(x-3) 2 & 12 f(3-x)+9 f(x)=3 x^2 & 12 f(3-x)+16 f(x)=4(x-3)^2 aligned aligned & 7 f(x)=4(x-3)^2=3 x^2 & = (4(24)^2-3.27^2 )- (4(22)^2-3.25^2 ) & =4 (24^2-22^2 )+3 (25^2-27^2 ) & =4(46)(2)+3(52)(-2) & =8 46 6 52 & f(27)-f(25)= 1 7 (46 8-6 52)= ( 56 7 )=8 aligned

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