Question

Consider the set \( S \) of points \( (x, y) \in \mathbb{R}^2 \) which minimize the real-valued function \( f(x, y) = (x + y - 1)^2 + (x + y)^2 \). Which of the following statements is true about the set \( S \)?

• A. The number of elements in the set \( S \) is finite and more than one.
• B. The number of elements in the set \( S \) is infinite.
• C. The set \( S \) is empty.
• D. The number of elements in the set \( S \) is exactly one.

Hint:

When minimizing functions of multiple variables, look for ways to reduce them to a function of a single variable by substitution. In this case, recognizing symmetry or common terms like \( x + y \) can greatly simplify the problem.

Answer 1

The Correct Option is B

We are given the function: \[ f(x, y) = (x + y - 1)^2 + (x + y)^2 \] Let \( z = x + y \). Then: \[ f(x, y) = (z - 1)^2 + z^2 = z^2 - 2z + 1 + z^2 = 2z^2 - 2z + 1 \] Now minimize: \[ g(z) = 2z^2 - 2z + 1 \] This is a quadratic function. The minimum occurs at: \[ z = \frac{-(-2)}{2 \cdot 2} = \frac{2}{4} = \frac{1}{2} \] So, the minimum value of the original function occurs when: \[ x + y = \frac{1}{2} \] All points \( (x, y) \in \mathbb{R}^2 \) such that \( x + y = \frac{1}{2} \) will minimize the function. This is a line in \( \mathbb{R}^2 \), and therefore the set \( S \) is infinite.