Introduction: In this problem, we are given a function f(x, y, z) = x^2yz and we need to find the greatest value of the directional derivative of this function at the point (1, 4, 1). The directional derivative measures the rate at which the function changes in a particular direction.

Step 1: Find the Gradient: To find the directional derivative, we first need to find the gradient of the function. The gradient is a vector that points in the direction of the greatest increase of the function.

The gradient of a function f(x, y, z) is given by:
∇f = ∂f/∂x * i + ∂f/∂y * j + ∂f/∂z * k

Here, i, j, and k are the unit vectors in the x, y, and z directions respectively.

For our function f(x, y, z) = x^2yz, let's find the partial derivatives:
∂f/∂x = 2xyz
∂f/∂y = x^2z
∂f/∂z = x^2y

Therefore, the gradient ∇f is given by:
∇f = (2xyz)i + (x^2z)j + (x^2y)k

Step 2: Find the Directional Derivative: The directional derivative of a function f(x, y, z) in the direction of a unit vector u = ai + bj + ck is given by the dot product of the gradient and the unit vector:
Df(u) = ∇f · u

In our case, we want to find the greatest value of the directional derivative. So, we need to find the unit vector that gives the maximum dot product with the gradient.

Step 3: Normalize the Gradient: To find the maximum dot product, we need to normalize the gradient vector. The normalized gradient vector is obtained by dividing the gradient vector by its magnitude.

The magnitude of the gradient vector is given by:
|∇f| = sqrt((2xyz)^2 + (x^2z)^2 + (x^2y)^2)

Step 4: Find the Unit Vector: Now, we can find the unit vector u that gives the maximum dot product with the normalized gradient. To do this, we divide the normalized gradient vector by its magnitude.

The unit vector u is given by:
u = (∇f / |∇f|)

Step 5: Calculate the Directional Derivative: Finally, we can calculate the directional derivative Df(u) by taking the dot product of the normalized gradient and the unit vector u.

Df(u) = (∇f / |∇f|) · (∇f / |∇f|)

Step 6: Substitute the Values: Now, we substitute the given point (1, 4, 1) into the expression for the directional derivative to find its greatest value.

Conclusion: After following these steps, we will obtain the greatest value of the directional derivative of the function f(x, y, z) = x^2yz at the point (1, 4, 1).