We are tasked with finding the points where the minimum rate of increase of the function $f(x, y) = x^2 + y^2 - 2x - 4y$ occurs in the direction of the vector $\mathbf{i} + \mathbf{j}$.

Steps:

1. Gradient of the function: The rate of increase in any direction is governed by the gradient of the function, $\nabla f(x, y)$. The gradient points in the direction of the maximum rate of increase, and is given by: $\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$ So, for $f(x, y) = x^2 + y^2 - 2x - 4y$, we find: $\frac{\partial f}{\partial x} = 2x - 2$ $\frac{\partial f}{\partial y} = 2y - 4$ Therefore, the gradient is: $\nabla f(x, y) = (2x - 2, 2y - 4)$

2. Direction of the vector $\mathbf{i} + \mathbf{j}$: The vector $\mathbf{i} + \mathbf{j}$ corresponds to the direction $\hat{u} = (1, 1)$. The rate of increase in this direction is given by the dot product: $\nabla f(x, y) \cdot \hat{u}$ This will give us the rate of change of the function in the direction of $\mathbf{i} + \mathbf{j}$.

3. Dot product of gradient and direction vector: $\nabla f(x, y) \cdot (1, 1) = (2x - 2)(1) + (2y - 4)(1) = 2x - 2 + 2y - 4 = 2x + 2y - 6$ We want to find the points where this rate of change is minimized. To find the critical points, set the expression equal to zero (since the minimum could occur where the rate of change is zero): $2x + 2y - 6 = 0$ Simplifying: $x + y = 3$

4. Critical points: The points that satisfy $x + y = 3$ lie along the line where this equation holds. Thus, any point on this line is a candidate for having the minimum rate of increase in the direction of $\mathbf{i} + \mathbf{j}$.

Conclusion:

The points where the minimum rate of increase of the function $f(x, y)$ in the direction of $\mathbf{i} + \mathbf{j}$ occurs are all the points that satisfy the equation: $x + y = 3$

Would you like any further clarification or details on this solution?

Here are five related questions for further practice:

1. What is the direction of maximum rate of increase of $f(x, y)$?
2. How do you find the rate of change of $f(x, y)$ in an arbitrary direction?
3. Can we determine the points where the maximum rate of increase occurs for this function?
4. What is the geometric interpretation of the gradient vector?
5. How would you solve this problem if the direction vector were different?

Tip: The gradient of a function always points in the direction of the steepest ascent, and its magnitude gives the rate of increase.