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`        let f(x+y/2)= f(x)+ f(y) / 2 for all real x and y . if f'(0) exists and equals -1 and f(0)=1,find f(2) ans is -1`
4 years ago

## Answers : (1) Arun Kumar
IIT Delhi
256 Points
```							Hif(x/2) = (1/2)f(x) + (1/2)f(0) = (1/2)f(x) + 1/2f(x) = (1/2)f(2x) + 1/2f ' (0) = lim(x->0) (f(x) - f(0))/xf ' (0) = lim(x->0) (f(x) - f(0))/xf ' (0) = lim(x->0) (f(2x) - 1)/(2x) = (1/2)lim(x->0) (f(2x) -1)/xf ' (x) = lim(y->0) ( f(x+y) - f(x) ) /yf(x+y) = f( (2x)/2 + (2y)/2).f(x+y) = (1/2)f(2x) + (1/2)f(2y)f '(x) = lim(y->0) [ (1/2)f(2x) + (1/2)f(2y) - f(x)]/y(1/2)f(2x) - f(x) = -1/2f ' (x) = lim(y->0) [ (1/2)f(2y) - 1/2] /y = (1/2) lim(y->0) [f(2y) - 1]/y(1/2) lim(y->0) [f(2y) - 1]/y = f'(0)f '(x) = -1, so f(x) = -x + c1 = f(0) = c so f(x) = 1 - xf(2) = 1 - 2 = -1Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
```
4 years ago
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