Question

"Use the fact that a continuous function f would take an absolute maximum and an absolute minimum on a bounded closed set (you can consider it to be a bounded set together with its boundary here). On the region R = (x, y) ∈ R² , which is not closed, consider the function f(x, y) = xy + x^(1/2) + y^(1/2). Find the minimum of f on R. Hint: Draw three curves: x = 1/4, y = 1/4, and xy = 4 on R."

Answer 1

To find the minimum of the function f(x, y) = xy + x^(1/2) + y^(1/2) on the region R = (x, y) ∈ R² , we can use the fact that a continuous function would take an absolute minimum on a bounded closed set.

Step-by-step explanation:

To find the minimum of the function f(x, y) = xy + x^(1/2) + y^(1/2) on the region R = x > 0, y > 0, we can use the fact that a continuous function would take an absolute minimum on a bounded closed set.

Although R is not closed, we can consider it as a bounded set together with its boundary. We can start by drawing three curves: x = 1/4, y = 1/4, and xy = 4 on R.

By analyzing the behavior of these curves and their intersections, we can identify potential points where the minimum might occur. Then, we can evaluate the function at these points and determine the absolute minimum. Let's proceed step by step:

Draw the curves x = 1/4, y = 1/4, and xy = 4 on the region R.

Identify the intersection points.

Evaluate the function f(x, y) at these intersection points.

Determine the minimum value of f(x, y) among the evaluated points.