1. **State the problem:** We are given the function $f(x,y) = y^2 + xy \ln x$ and need to understand or analyze it. 2. **Recall the components:** The function involves variables $x$ and $y$, with $y^2$ being a simple quadratic term in $y$, and $xy \ln x$ involving a product of $x$, $y$, and the natural logarithm of $x$. 3. **Important notes:** - The natural logarithm $\ln x$ is defined only for $x > 0$. - The function is defined for $x > 0$ and any real $y$. 4. **Intermediate work:** - The function can be written as $$f(x,y) = y^2 + xy \ln x.$$ - No further simplification is possible without additional context (e.g., finding partial derivatives, critical points, or evaluating at specific points). 5. **Summary:** The function $f(x,y)$ combines a quadratic term in $y$ and a product term involving $x$, $y$, and $\ln x$. It is defined for $x > 0$. Final answer: $$f(x,y) = y^2 + xy \ln x.$$