Final answer:
The gradient of the function f(x, y, z) = ln(x² + 4y² + 3z²) at the point (4, 4, 4) is (1/6, 2/3, 1/2).
Step-by-step explanation:
The gradient of a function is a vector that points in the direction of the greatest rate of change of the function. To compute the gradient of the function f(x, y, z) = ln(x² + 4y² + 3z²), we need to find the partial derivatives of the function with respect to x, y, and z. Then, we evaluate those partial derivatives at the given point (4, 4, 4).
The partial derivative of f with respect to x is 2x / (x² + 4y² + 3z²). The partial derivative of f with respect to y is 8y / (x² + 4y² + 3z²). The partial derivative of f with respect to z is 6z / (x² + 4y² + 3z²).
Evaluating the partial derivatives at (4, 4, 4), we have: ∂f/∂x = 8/(16 + 64 + 48) = 1/6, ∂f/∂y = 32/(16 + 64 + 48) = 2/3, and ∂f/∂z = 24/(16 + 64 + 48) = 1/2.
Therefore, the gradient of the function at the point (4, 4, 4) is (1/6, 2/3, 1/2).