Let f: R R be a differentiable function such that f ( x+y 3 ) = f(x) + f(y) 3 for all x, y R , and f'(0) = 3 . Then the minimum value of the function g(x) = 3 + e^x f(x) , is:

Step-by-step solution

Given f ( x+y 3 ) = f(x) + f(y) 3 Substituting x = 0 and y = 0 , we get: f(0) = 2f(0) 3 f(0) = 0 Differentiating the given equation partially with respect to x , treating y as a constant: f' ( x+y 3 ) 1 3 = f'(x) 3 f' ( x+y 3 ) = f'(x) Substituting x = 0 , we get: f' ( y 3 ) = f'(0) = 3 Since this is true for all y R , f'(x) = 3 for all x R . Integrating both sides with respect to x : f(x) = 3x + C Using f(0) = 0 , we get C = 0 . Thus, f(x) = 3x . Now, the function g(x) is given by: g(x) = 3 + e^x f(x) = 3 + 3x e^x