Use Lagrange's Multipliers to find all points on the ellipse: $8 x^{2}+2 y^{2}=16$ for which the product $P=x y$ is a minimum.

Step 1 ：Find the gradient of P: \(\nabla P = \begin{bmatrix} y \\ x \end{bmatrix}\)

Step 2 ：Find the gradient of f: \(\nabla f = \begin{bmatrix} 16x \\ 4y \end{bmatrix}\)

Step 3 ：Set up the Lagrange multiplier equation: \(\begin{bmatrix} y \\ x \end{bmatrix} = \lambda \begin{bmatrix} 16x \\ 4y \end{bmatrix}\)

Step 4 ：Solve the system of equations: \(y = 16\lambda x\), \(x = 4\lambda y\), and \(8x^2 + 2y^2 = 16\)

Step 5 ：Consider the case when \(\lambda = \frac{1}{8}\):

Step 6 ：Substitute \(\lambda = \frac{1}{8}\) into the equations to find the values of \(x\) and \(y\):

Step 7 ：When \(y = 2\), \(x = 1\)

Step 8 ：When \(y = -2\), \(x = -1\)

Step 9 ：Consider the case when \(\lambda = -\frac{1}{8}\):

Step 10 ：Substitute \(\lambda = -\frac{1}{8}\) into the equations to find the values of \(x\) and \(y\):

Step 11 ：When \(y = 2\), \(x = -1\)

Step 12 ：When \(y = -2\), \(x = 1\)

Step 13 ：The points on the ellipse for which the product \(P = xy\) is a minimum are \((1, 2)\), \((-1, -2)\), \((-1, 2)\), and \((1, -2)\)