Final answer:
A C^2 function f(x, y) with the given partial derivatives fx = 2x - 5y and fy = 4x + y cannot exist, because the mixed partial derivatives are not equal, which fails to satisfy the Clairaut's theorem.
Step-by-step explanation:
The question asks if a function f(x, y) that is twice differentiable (C^2 function) can exist with partial derivatives fx = 2x - 5y and fy = 4x + y. To determine if such a function exists, we must check if the mixed partial derivatives of f are equal, that is, if the Clairaut's theorem (also known as the Schwarz theorem) holds, which states that for a C^2 function, the mixed partial derivatives are equal no matter the order of differentiation.
For this to be true, the partial derivative of fx with respect to y must be equal to the partial derivative of fy with respect to x. By differentiating fx with respect to y, we get -5. Similarly, by differentiating fy with respect to x, we obtain 4. Since these two values are not equal, the equality of mixed partials does not hold, and therefore, a C^2 function with the given partial derivatives cannot exist.