Question

Let f (x, y)=(x²+y²)⋅(x⋅y+1),x,y∈R.

(a) Find the first- and second-order partial derivatives of f.
(b) Find all stationary points of f.
(c) Classify the stationary points.

Answer 1

To find the first- and second-order partial derivatives of f(x, y) = (x²+y²)⋅(x⋅y+1), differentiate with respect to x and y separately. Then, differentiate the first-order partial derivatives with respect to x and y to find the second-order partial derivatives.

Step-by-step explanation:

To find the first-order partial derivatives of f(x, y) = (x²+y²)⋅(x⋅y+1), we differentiate with respect to x and y separately:

∂f/∂x = (2x⋅(x²+y²) + (x⋅y+1)⋅2x)

∂f/∂y = (2y⋅(x²+y²) + (x⋅y+1)⋅2y)

To find the second-order partial derivatives, we differentiate the first-order partial derivatives with respect to x and y:

∂²f/∂x² = 2(x²+y²) + 4x² + (x⋅y+1)⋅2

∂²f/∂y² = 2(x²+y²) + 4y² + (x⋅y+1)⋅2

∂²f/∂x∂y = 2xy + 2x + 2xy + 2y = 4xy + 2x + 2y