If g(x) = (x^2 + 2x + 1) f(x) such that f(0) = 5 and _ x 0 f(x)-5 x = 4 then g'(0) =

Step-by-step solution

Given g(x) = (x^2 + 2x + 1) f(x) Differentiating with respect to x using the product rule: g'(x) = (2x + 2) f(x) + (x^2 + 2x + 1) f'(x) Substituting x = 0 : g'(0) = 2 f(0) + f'(0) From the given limit, _ x 0 f(x)-5 x = 4 , which represents the derivative of f(x) at x = 0 . Thus, f'(0) = 4 . Also given f(0) = 5 . Substituting these values: g'(0) = 2(5) + 4 = 14 Answer: 14