Final answer:
The maximum rate of change of f at the point (2, 4) is 64 in the direction of the vector (64, 64).
Step-by-step explanation:
The maximum rate of change of f at the point (2, 4) and the direction in which it occurs can be found by taking the partial derivatives of f(x, y) = 4y² * x with respect to x and y.
First, take the partial derivative with respect to x:
∂f/∂x = 4y²
Next, take the partial derivative with respect to y:
∂f/∂y = 8xy
Substituting the values x = 2 and y = 4 into these equations, we get:
∂f/∂x = 4(4)² = 64
∂f/∂y = 8(2)(4) = 64
Therefore, the maximum rate of change of f at the point (2, 4) is 64 in the direction of the vector (64, 64).