Final answer:
To find the maximum rate of change of f at the given point (4, 9), we need to find the gradient vector of f(x, y) and calculate its magnitude. The maximum rate of change is approximately 39.39, and it occurs in the direction of the vector ⟨2.031, 0.901⟩.
Step-by-step explanation:
To find the maximum rate of change of f at the given point, we need to find the gradient vector of f(x, y) at (4, 9). The gradient vector is ⟨∂f/∂x, ∂f/∂y⟩. Given that f(x, y) = 4xy, the partial derivatives are ∂f/∂x = 4y and ∂f/∂y = 4x. So, the gradient vector at (4, 9) is ⟨4(9), 4(4)⟩ which simplifies to ⟨36, 16⟩.
The maximum rate of change is the magnitude of the gradient vector, which can be calculated using the formula √(a² + b²). In this case, the maximum rate of change is √(36² + 16²), which simplifies to √(1296 + 256) = √1552 ≈ 39.39.
The direction in which the maximum rate of change occurs is given by the direction vector of the gradient vector, which is the vector ⟨4(9), 4(4)⟩ divided by its magnitude. So, the direction vector is ⟨36/√1552, 16/√1552⟩ = ⟨9√2/√388, 4√2/√388⟩ ≈ ⟨2.031, 0.901⟩.