Final answer:
To find the four second partial derivatives of the function z = 8xe^y - 16ye^-x, calculate the derivative of first partial derivatives with respect to x and y, resulting in the four second partials including the mixed ones. By Clairaut's theorem, mixed partial derivatives should be equal if certain conditions of continuity are met.
Step-by-step explanation:
The question asks to find the four second partial derivatives of the function z = 8xey − 16ye−x and observe that the second mixed partials are equal. The second partial derivatives are found by first taking the partial derivative of z with respect to x and y, and then differentiating a second time with respect to each variable separately.
The second partial derivative of z with respect to x twice (denoted as ∂2z/∂x2) is the derivative of the first partial derivative of z with respect to x. Likewise, the second partial derivative of z with respect to y twice (∂2z/∂y2) is the derivative of the first partial derivative of z with respect to y. The mixed partial derivatives (∂2z/∂x∂y and ∂2z/∂y∂x) involve taking the partial derivative of z first with respect to x then y, and vice versa.
By Clairaut's theorem, if the function and its partial derivatives are continuous, the mixed partial derivatives (∂2z/∂x∂y and ∂2z/∂y∂x) will be equal. We can calculate these derivatives for the given function to demonstrate this property.