Aerospace Wing Profile Lift-to-Drag Optimization (Aerospace & Mechanical Engineering)

An aerospace engineer at the UNLV Howard R. Hughes College of Engineering is prototyping a high-efficiency transonic wing section for an autonomous aerial survey drone. Aerodynamic wind-tunnel testing reveals that the non-dimensional lift-to-drag efficiency ratio $E(x, y)$ depends on the wing's angle of attack $x$ (in degrees, centered around nominal cruise trim) and the maximum camber percentage $y$ (in percent of chord length).

The empirical efficiency performance surface is modeled across the operational flight envelope by the unconstrained multivariable cubic function:

1. Determine the first-order partial derivatives $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ to formulate the gradient vector $\nabla E(x, y)$.
2. Locate all critical points $(x_0, y_0)$ where the aerodynamic performance gradient vanishes ($\nabla E = \mathbf{0}$).
3. Compute all second-order partial derivatives and formulate the Hessian discriminant $D(x, y) = E_{xx}E_{yy} - (E_{xy})^2$.
4. Apply the Second Derivative Test to classify each critical point, identifying which configuration represents a stable local maximum in cruise efficiency versus an unstable saddle state.
5. Evaluate the maximum aerodynamic efficiency ratio $E_{\max}$ at the optimal design point.