Question

if f(x+y/2)=f(x)+f(y)/2, f'(0)=a and f'(0)=b then find f"(x) where y is independent of x.​

Answer 1

Answer:

Hi

f(x/2) = (1/2)f(x) + (1/2)f(0) = (1/2)f(x) + 1/2

f(x) = (1/2)f(2x) + 1/2

f ' (0) = lim(x->0) (f(x) - f(0))/x

f ' (0) = lim(x->0) (f(x) - f(0))/x

f ' (0) = lim(x->0) (f(2x) - 1)/(2x) = (1/2)lim(x->0) (f(2x) -1)/x

f ' (x) = lim(y->0) ( f(x+y) - f(x) ) /y

f(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) = -1/2

f ' (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 + c

1 = f(0) = c so f(x) = 1 - x

f(2) = 1 - 2 = -1

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty

Step-by-step explanation: