If x^2 e^x dx = e^x f(x) + c , then the minimum value of f(x) is ...

Step-by-step solution

Using integration by parts: x^2 e^x dx = x^2 e^x - 2x e^x dx = x^2 e^x - 2(x e^x - e^x dx) = x^2 e^x - 2x e^x + 2e^x + c = e^x (x^2 - 2x + 2) + c Comparing with the given expression, we get f(x) = x^2 - 2x + 2 . To find the minimum value of f(x) , we can complete the square: f(x) = (x - 1)^2 + 1 Since (x - 1)^2 0 for all real x , the minimum value of f(x) is 1 . Answer: 1