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Let f(x+y /2 )=f(x)+f(y) /2 for all values of x and y belongs to R. If f '(0) exists and equals -1 and f(0)=1, find f(2)? ans is -1. plz explains the steps.

Let f(x+y /2 )=f(x)+f(y) /2 for all values of x and y belongs to R. If f'(0) exists and equals -1 and f(0)=1, find f(2)?


 ans is -1.     plz explains the steps.

Grade:12

1 Answers

saiomkar kandukuri
33 Points
9 years ago

differentiate wrt x then we will get 

df(x+ y/2)=df(x).............(1)

differentiate the question wrt to y

df(x+ y/2)*(1/2) =(1/2)df(y)

divide the two equations

we get df(y)=df(x)*2=k,......................(3)

   kis a constant

integrate with and find f(x)

f(x)=kx/2

given f''(x)=-1=k/2

so the value of k is known ......hence function is known     ....find the value of f(2)

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