Let f ( 2 x + y ) = 2 1 ( f ( x ) + f ( y )) for real x and y. If f ′ ( 0 ) exists and equals to -1 and f ( 0 ) = 1 then the value of f ( 2 ) is

Question

Let f ( 2 x + y ​ ) = 2 1 ​ ( f ( x ) + f ( y )) for real x and y. If f ′ ( 0 ) exists and equals to -1 and f ( 0 ) = 1 then the value of f ( 2 ) is

TestHub · Accepted Answer

f ′ ( x ) f ′ ( x ) ​ = h → 0 lim ​ h f ( x + h ) − f ( x ) ​ = h → 0 lim ​ h 2 f ( 2 x ) + f ( 2 h ) ​ − f ( x ) ​ = − 1 ; f ( 2 x ) = 2 f ( x ) − 1 ⇒ f ( x ) = 1 − x ​

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