Final answer:
To prove that f(x,y)=xcos(x+y) is of class C∞, we need to show the differentiability, continuity of partial derivatives, existence and continuity of mixed partial derivatives, and convergence of the Taylor series expansion. The partial derivatives ∂f/∂x and ∂f/∂y exist and are continuous. All mixed partial derivatives of f(x,y) exist and are continuous. The Taylor series expansion of f(x,y) converges for all x and y.
Step-by-step explanation:
To show that the function f(x,y)=xcos(x+y) is of class C∞, we need to prove the differentiability, continuity of partial derivatives, existence and continuity of mixed partial derivatives, and convergence of the Taylor series expansion.
a. To prove the differentiability of f(x,y), we need to show that the partial derivatives ∂f/∂x and ∂f/∂y exist and are continuous. In this case, ∂f/∂x = cos(x+y) - x sin(x+y) and ∂f/∂y = -x sin(x+y). Both of these derivatives are continuous.
b. To demonstrate the continuity of the partial derivatives of f(x,y), we already showed in part a that both ∂f/∂x and ∂f/∂y are continuous.
c. To show that all mixed partial derivatives of f(x,y) exist and are continuous, we need to compute the second partial derivatives. We find that the second partial derivatives ∂2f/∂x2 = -2sin(x+y) and ∂2f/∂y2 = -x2cos(x+y). Both of these derivatives exist and are continuous.
d. To confirm the convergence of the Taylor series expansion of f(x,y) for all x and y, we can use the Taylor series expansion formula. This formula states that if a function is infinitely differentiable, its Taylor series expansion converges to the function itself. Since we have shown that f(x,y) is of class C∞, we can conclude that the Taylor series expansion of f(x,y) converges for all x and y.