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

Mathematics - Functions Question with Solution | TestHub

Options:

Doubts & Discussion