Final answer:
To find fₓ and fᵧ for the function f(x, y) = e^(13xy), we differentiate with respect to x and y, treating the other variable as a constant. Then, we set up the Lagrangian function using the method of Lagrange multipliers and find the partial derivatives with respect to x, y, and λ. Finally, evaluate f(x, y) at the critical point to find the minimum value of x + y and the maximum value of xy.
Step-by-step explanation:
The function is given as f(x,y) = e^(13xy). To find the partial derivative fₓ, we differentiate f(x,y) with respect to x while treating y as a constant. Similarly, to find the partial derivative fᵧ, we differentiate f(x,y) with respect to y while treating x as a constant.
To apply the method of Lagrange multipliers, we set up the Lagrangian function L(x, y, λ) = f(x, y) - λ(g₁(x, y) - c₁) - λ(g₂(x, y) - c₂), where g₁(x, y) = xy - 324 and g₂(x, y) = x + y - 324 are the constraints. We then find the partial derivatives of L(x, y, λ) with respect to x, y, and λ, and set them equal to zero to solve for x, y, and λ.
The minimum value of x + y can be found by evaluating f(x, y) at the critical point obtained from the Lagrange multipliers method. Similarly, the maximum value of xy can be found by evaluating f(x, y) at the critical point.