f(x₁,x₂)=(x₁−x₂)² and g(x₁,x₂)=2x₁²+2x₂²−x₁x₂. S1: f⁻¹(K) is compact for every compact K⊂ℝ. S2: g⁻¹(K) is compact for every such K. Choose the truth values.
For K={0}, f⁻¹(K) is the unbounded diagonal line, so S1 fails. For g, $g\ge\frac32(x_1^2+x_2^2)$, making the inverse image of a bounded K bounded. It is also closed by continuity; Heine–Borel makes it compact. **S2 only, B**.
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