The extremes sit on the boundary

Let u = u(x, y) be the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ on (0, 1) × (0, 1), with $u(x, 0) = e^(\pi{}x), u(x, 1) = -e^(\pi{}x)$ for $x \in [0, 1], u(0, y) = \cos(\pi{}y) + \sin(\pi{}y)$ and $u(1, y) = e^\pi(\cos(\pi{}y) + \sin(\pi{}y))$ for y ∈ [0, 1]. Then there exists a point $(x_{0}, y_{0}) \in (0, 1) \times (0, 1)$ such that