Final answer:
To find the maximum rate of change of f at the given point (3, 9), we need to calculate the gradient of the function at that point. The gradient is (-15, 30), which represents the direction in which the maximum rate of change occurs. The maximum rate of change is approximately 33.54 and the direction is (-0.447, 0.894).
Step-by-step explanation:
To find the maximum rate of change of f at a given point, we need to calculate the gradient of the function at that point. The gradient of a function represents the rate of change of the function with respect to its variables. In this case, we have the function f(x, y) = 5y²/x. To calculate the partial derivatives, we differentiate the function with respect to x and y separately.
First, we find the partial derivative of f with respect to x:
∂f/∂x = -(5y²)/x²
Next, we find the partial derivative of f with respect to y:
∂f/∂y = 10y/x
Using these partial derivatives, we can find the gradient of f at the point (3, 9). Substituting x = 3 and y = 9 into the derivatives:
∂f/∂x = -(5(9)²)/(3)² = -135/9 = -15
∂f/∂y = 10(9)/(3) = 30
Therefore, the gradient of f at the point (3, 9) is (-15, 30). The maximum rate of change of f at this point is the magnitude of the gradient vector, which is √((-15)² + 30²) = √(225 + 900) = √1125 ≈ 33.54. The direction in which the maximum rate of change occurs is given by the direction vector of the gradient, which is (-15/33.54, 30/33.54) ≈ (-0.447, 0.894).