Final answer:
To find the direction in which the function f(x, y) = x³y - x²y⁴ decreases fastest at the point (2, -3), we need to find the gradient vector of the function at that point.
Step-by-step explanation:
To find the direction in which the function f(x, y) = x³y - x²y⁴ decreases fastest at the point (2, -3), we need to find the gradient vector of the function at that point. The gradient vector points in the direction of the steepest increase of the function. However, we are looking for the direction of the steepest decrease, so we need to find the negative of the gradient vector.
Step 1: Calculate the partial derivative of the function with respect to x and y:
∂f/∂x = 3x²y - 2xy⁴
∂f/∂y = x³ - 4x²y³
Step 2: Plug in the values of x = 2 and y = -3 into the partial derivatives to get the gradient vector:
∇f(2, -3) = (3(2)²(-3) - 2(2)(-3)⁴, (2)³ - 4(2)²(-3)³) = (-108, 50)
Step 3: Find the negative of the gradient vector to get the direction of steepest decrease:
Direction of steepest decrease = -(-108, 50) = (108, -50)