Let `f((x+y)/2)=(f(x)+f(y))/2fora l lr e a lxa n dy` If `f^(prime)(0)` exists and equals `-1a n df(0)=1,t h e n f i n d f(2)dot`

Let f(x^m y^n)=mf(x)+nf(y) for all x , y in R^+ and for all m ,n in Rdot If f^(prime)(x) exists and has the value e/x , then find (lim)_(xvec0)(f(1+x))/x

If f((x+2y)/3)=(f(x)+2f(y))/3AAx ,y in Ra n df^(prime)(0)=1,f(0)=2, then find f(x)dot

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then find the value of f(7)dot

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

Let f be a function such that f(x+y)=f(x)+f(y) for all xa n dya n df(x)=(2x^2+3x)g(x) for all x , where g(x) is continuous and g(0)=3. Then find f^(prime)(x)dot

If f: R^+vecR ,f(x)+3xf(1/x)=2(x+1),t h e nfin df(x)dot

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real x and ydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then a) f(x) is positive for all real x b) f(x) is negative for all real x c) f(x)=0 has real roots d) Nothing can be said about the sign of f(x)

Let f(x y)=f(x)f(y)AAx , y in Ra n df is differentiable at x=1 such that f^(prime)(1)=1. Also, f(1)!=0,f(2)=3. Then find f^(prime)(2)dot

If f(x)=(f(x))/y+(f(y))/x holds for all real x and y greater than 0a n df(x) is a differentiable function for all x >0 such that f(e)=1/e ,t h e nfin df(x)dot

Let f(x+y)=f(x)+f(y)+2x y-1 for all real xa n dy and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, then prove that f(x)>0AAx in Rdot