Question

You and your colleagues are searching for an optimal point within your design space given by the objective function: F(x, y) = sin 2θ + x^4/2 + x^2y^ - 4 cosθ. You believe you've located a maximum point at (0.5,0.4). Is this point indeed a maximum? Why or why not? Mathematically justify your answer. (hint: Use the Hessian approach)

Answer 1

To determine whether the point (0.5, 0.4) is a maximum, we need to examine the Hessian matrix of F(x, y) at that point. The Hessian matrix is given by:

H = [∂²F/∂x² ∂²F/∂x∂y]
[∂²F/∂y∂x ∂²F/∂y²]

Evaluating the partial derivatives of F(x, y) and plugging in (0.5, 0.4) gives:

F(0.5, 0.4) = sin(2θ) + 0.34375 - 3.2cos(θ)
∂F/∂x = 2x^3 + xy^2
∂F/∂y = x^2y - 4cos(θ)
∂²F/∂x² = 6x^2 + y^2
∂²F/∂y² = x^2
∂²F/∂x∂y = 2xy

Plugging in (0.5, 0.4) gives:

∂F/∂x = 0.5
∂F/∂y = -3.2
∂²F/∂x² = 1.2
∂²F/∂y² = 0.25
∂²F/∂x∂y = 0.4

Therefore, the Hessian matrix at (0.5, 0.4) is:

H = [1.2 0.4]
[0.4 0.25]

To determine whether this is a maximum or minimum, we need to examine the eigenvalues of the Hessian matrix. The eigenvalues are given by the roots of the characteristic equation:

det(H - λI) = 0

where I is the identity matrix. Plugging in the Hessian matrix and solving for λ gives:

det(H - λI) = (1.2 - λ)(0.25 - λ) - 0.16 = λ^2 - 1.45λ + 0.08 = 0

Solving for the roots of this quadratic equation gives:

λ1 ≈ 1.37
λ2 ≈ 0.08

Since both