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

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, the prove that f(x)>0AAx in Rdot

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, the prove that f(x)>0AAx in Rdot

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, the prove that f(x)>0AAx in Rdot

A function f: R->R satisfies the equation f(x+y)=f(x)f(y) for all x , y in R and f(x)!=0 for all x in Rdot If f(x) is differentiable at x=0a n df^(prime)(0)=2, then prove that f^(prime)(x)=2f(x)dot

A function f: RvecR satisfies the equation f(x+y)=f(x)f(y) for all x , y in Ra n df(x)!=0fora l lx in Rdot If f(x) is differentiable at x=0a n df^(prime)(0)=2, then prove that f^(prime)(x)=2f(x)dot

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot 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 f(x) is positive for all real x f(x) is negative for all real x f(x)=0 has real roots Nothing can be said about the sign of f(x)

If f(xy)=(f(x))/y+(f(y))/x holds for all real x and y greater than 0 and f(x) is a differentiable function for all x >0 such that f(e)=1/e , then find f(x)

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 ,then find f(x)dot

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)

Let f(x+y) = f(x) + f(y) - 2xy - 1 for all x and y. If f'(0) exists and f'(0) = - sin alpha , then f{f'(0)} is