Question

Let
𝑆 = x\left\\{(𝑥, 𝑦) ∈ ℝ^2 : 𝑥 > 0, 𝑦 > 0\right\\} ,
and f: S → ℝ be given by
$f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.$
Then, which of the following statements is/are TRUE ?

• A. There is a unique point in S at which f(x, y) attains a local maximum
• B. There is a unique point in S at which f(x, y) attains a local minimum
• C. For each point (x0, y0) ∈ S, the set {(x, y) ∈ S : f(x, y) = f(x0, y0) } is bounded
• D. For each point (x0, y0) ∈ S, the set {(x, y) ∈ S : f(x, y) = f(x0, y0) } is unbounded

Answer 1

Answer: (B), (C)

There is a unique point in S at which f(x, y) attains a local minimum

Answer 2

The correct option is (B) : There is a unique point in S at which f(x, y) attains a local minimum and (C) : For each point (x0, y0) ∈ S, the set {(x, y) ∈ S : f(x, y) = f(x0, y0) } is bounded.