Problem

The key is to recognize that $g'(x) = \frac{2}{x + g(x)}$ creates a relationship between how fast $g$ grows and its current value. Rearrange to $(x + g(x)) g'(x) = 2$, which is the derivative of $x \cdot g(x) + \frac{1}{2}g(x)^2$—a perfect setup for integration. By integrating both sides from 1 to 6 and using the initial condition $g(1) = 0$, you can solve for the maximum possible value of $g(6)$.

The key is to recognize that $g'(x) = \frac{2}{x + g(x)}$ creates a relationship between how fast $g$ grows and its current value. Rearrange to $(x + g(x)) g'(x) = 2$, which is the derivative of $x \cdot g(x) + \frac{1}{2}g(x)^2$—a perfect setup for integration. By integrating both sides from 1 to 6 and using the initial condition $g(1) = 0$, you can solve for the maximum possible value of $g(6)$.