The problem you're tackling involves the gradient of the temperature function $T(x, y, z)$. The gradient will help us find the rate of change in various directions and the direction of the maximum temperature increase.

Let's go through each part step by step:

1. Find the rate of change of the temperature at the point $(2, 1, 2)$ in the direction toward the point $(3, 3, 3)$

The rate of change of temperature in a given direction is given by the directional derivative, which is calculated using the gradient of the temperature function and the unit vector in the direction of the point.

Steps:

• Find the gradient of $T(x, y, z)$: $\nabla T(x, y, z)$.
• Find the unit vector in the direction from $(2, 1, 2)$ to $(3, 3, 3)$.
• Compute the directional derivative using the formula: $D_{\mathbf{u}} T = \nabla T(x, y, z) \cdot \mathbf{u}$ where $\mathbf{u}$ is the unit vector in the given direction.

2. In what direction does the temperature increase fastest at $(2, 1, 2)$?

The direction of fastest temperature increase is always in the direction of the gradient vector, $\nabla T(x, y, z)$, evaluated at the point $(2, 1, 2)$.

3. Find the maximum rate of increase at $(2, 1, 2)$

The maximum rate of increase is the magnitude of the gradient at the point $(2, 1, 2)$, which is given by: $\text{Maximum rate of increase} = |\nabla T(2, 1, 2)|$

Could you please provide the temperature function $T(x, y, z)$ so we can compute these values accurately?