GATE Electrical 2025 | Question: 1

Question

Consider the set $S$ of points $(x, y) \in \mathfrak{R}^{2}$ 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.

Answer 1

The function to minimize is f(x,y) = (x+y−1)^2 + (x+y)^2. 

Let u=x+y. Then the function can be written as g(u) = (u−1)^2 + u^2 

To minimize g(u), its derivative with respect to u and set it to 0:

g′(u)=2(u−1)+2u=2u−2+2u=4u−2

g′(u)=0

4u−2=0, so

u=1/2. The minimum value of the function occurs when

x+y=1/2

This equation represents a straight line in the plane, which contains an infinite number of points. 

(B) Therefore, the set S has an infinite number of elements.