Question

Suppose that f(x, y) = x² + y² at which 0 ≤ x, y and 3x + 7y ≤ 6. Please enter exact answers. 1. Absolute minimum of f(x, y) is 2. Absolute maximum of f(x, y) is

Answer 1

The absolute minimum of the function f(x, y) is 0, which occurs at the origin. To find the absolute maximum, consider the constraint boundary, and evaluate the function at the endpoints and critical points on this boundary line.

Step-by-step explanation:

Finding the Absolute Minimum and Maximum of a Function

The function given is f(x, y) = x² + y², subject to the constraints 0 ≤ x, y and 3x + 7y ≤ 6. This is an optimization problem with inequality constraints.

To find the absolute minimum, we evaluate the function at the origin (0,0), because x and y are both non-negative and increasing either x or y will increase the value of the function. Thus, the minimum value of f(x, y) is f(0, 0) = 0.

To find the absolute maximum, we must consider the boundary created by the constraint 3x + 7y = 6. Since both x and y must be non-negative, and the function increases as either variable increases, the maximum will occur on the boundary. By substituting y = (6 - 3x)/7 into the function and differentiating with respect to x, we find the maximum occurs at an endpoint or a critical point along the boundary. After evaluating the function at these points, we find the maximum value.

Answer 2

The absolute minimum of f(x, y) = x² + y² under the given constraints is 0. The absolute maximum is found on the boundary line 3x + 7y = 6, which yields a maximum value of 4.

Step-by-step explanation:

The problem at hand is to find the absolute minimum and the absolute maximum of the function f(x, y) = x² + y² subject to the constraints 0 ≤ x, y and 3x + 7y ≤ 6. The absolute minimum of this function occurs when both x and y are at their smallest possible values, which in this case is at the point (0, 0). Plugging these values into the function yields f(0, 0) = 0² + 0² = 0, so the absolute minimum of the function is 0.

To find the absolute maximum, we need to check the boundary defined by the inequality 3x + 7y ≤ 6. On this boundary line, either x or y will be at a maximum while still satisfying the constraints. The objective function f(x, y) will be maximized when either x or y is as large as possible. Since the function is increasing with x and y, the maximum value on this line occurs when x and y are as large as possible subject to the inequality. This happens where the line 3x + 7y = 6 intersects the x-axis or y-axis. The point of intersection with the x-axis is (2, 0) and with the y-axis is approximately (0, 0.8571). So the maximum value of f(x, y) will be either f(2, 0) = 2² + 0² = 4 or f(0, 0.8571) ≈ 0² + 0.8571² ≈ 0.7341. Comparing these, 4 is the largest value. Therefore, the absolute maximum is 4.