\[ g(x) = x \\ g'(x) = 1 \\ h(x)*g(x) = 500 \\ h(x) = \frac{500}{x} \\ h'(x) = -\frac{500}{x^2} \\ f(x) = g(x)+h(x) \\ f'(x) = 1 - \frac{500}{x^2} \\ \text{for minimum}\space f'(x) = 0\\ 0 = 1 - \frac{500}{x^2} \\ 1 =\frac{500}{x^2} \\ x^2 = 500 \\ x = \pm \sqrt{500} \\ \text{The negative roots will give the lowest sum.} \\ -\sqrt{500} + - \sqrt{500} = -44.72136 \]