Let $A = [0,1]^2$ and $f(x,y) = (e^x cos y, e^x sin y)$. Using the Mean Value Theorem, determine the maximum value of $||f(x,y)||$ where $(x,y) \in A$. Explain why the maximum is attained on A.
I have concluded that the maximum is attained on A because it is a bounded and closed set, and therefore compact. Since it is compact and the function is continuous, the maximum value is achieved on A. I have calculated the norm $||f(x,y)|| = \sqrt{(e^x \cos y)^2 + (e^x \sin y)^2}$ and found it to be equal to $e^x$. I don't understand how to apply the Mean Value Theorem here, and I need assistance with that.