Choosing an Event Location Based on Elevation and Differentiability: Farmer McLauren's Suggestions

Question

\begin{align*} &\text{Laprox, MB is excited to host the Differentiable Music Festival in the countryside surrounding the town. Farmer McLauren has offered his farmland as a place to construct the stage and seating area, but his farm is, in effect, a mountain. As such, it is important to build on a relatively flat area of land. Farmer McLauren’s farm is vast, almost nine thousand acres in size, occupying a square six kilometers to a side. Surveyors working in advance to scout the best location have modeled his farmland’s elevation with the following equation:} \\ &z(x, y) = 2 - 0.3x - 0.4\sqrt{x^2 + y^2} + 0.2\cos(\pi x) - 0.1y^2 \text{ where } x, y \text{ are position in kilometers relative to his farmhouse, and } z \text{ is the elevation in kilometers relative to the town square} \\ & \\ &\text{Farmer McLauren’s first suggestion is to hold the festival around his house, using his roof as the stage. Here the function } z \text{ is not differentiable at the origin and there is no tangent plane at }(0,0). \\ & \\ &\text{Farmer McLauren also thinks it might be a good idea to try holding the concert by the silo 2 km west of his house. Here the function is differentiable at } (-2,0) \text{ and the tangent plane is given by } z = 1 - \frac{x}{2}. \\ & \\ &\text{Farmer McLauren has one final suggestion: the “field where the goats graze,” 2 km south of his house. Here the function is differentiable at } (0,-2) \text{ and the tangent plane is given by } z = -\frac{x}{2} + \frac{4y}{5} + \frac{13}{5} \\ & \\ &\text{Which of Farmer McLauren’s suggestions do you think should be followed, and why?} \end{align*}