If `f((x+2y)/3)=(f(x)+2f(y))/3\ AA\ x ,\ y in R` and `f^(prime)(0)=1` , `f(0)=2` , then find `f(x)` .

If f((x+2y)/(3))=(f(x)+2f(y))/(3)AA x,y in R and f'(0)=1,f(0)=2 then find f(x) .

If f((x + 2y)/3)=(f(x) + 2f(y))/3 AA x, y in R and f(x) is continuous at x = 0. Prove that f(x) is continuous for all x in R.

(i) if f ( (x + 2y)/3)]= (f(x) +2f(y))/3 , " x, y in R and f (0) exists and is finite ,show that f(x) is continuous on the entire number line. (ii) Let f : R to R satisfying f(x) -f (y) | le | x-y|^(3) , AA x, y in R , The prove that f(x) is a constant function.

Let f(x+y)+f(x-y)=2f(x)f(y)AA x,y in R and f(0)=k, then

Let f(x + y) = f(x) .f(y) AA x, y in R suppose that f(k) =3, k in R and f'(0) = 11 then find f'(k)

If a function 'f' satisfies the relation f(x)f^('')(x)-f(x)f^(')(x) -f^(')(x)^(2)=0 AA x in R and f(0)=1=f^(')(0) . Then find f(x) .

If f((x)/(y))=(f(x))/(f(y)),AA y,f(y)!=0 and f'(1)=2, find f(x)

If f(x+y) = f(x) + f(y) + |x|y+xy^(2),AA x, y in R and f'(0) = 0 , then