Discussion
The set
\[ S = \{(x_{1}, x_{2}) : 3x_{1}^{2} + 2x_{2}^{2} \le 6\} \]
is a :
Answer: C
S is a convex set.
S is the solid ellipse \(\dfrac{x_{1}^{2}}{2} + \dfrac{x_{2}^{2}}{3} \le 1\). The function \(g(x_{1}, x_{2}) = 3x_{1}^{2} + 2x_{2}^{2}\) is convex (a sum of convex quadratics), and a sublevel set \(\{g \le 6\}\) of a convex function is convex: if \(g(p) \le 6\) and \(g(q) \le 6\), then \(g(\lambda p + (1 - \lambda)q) \le \lambda g(p) + (1 - \lambda)g(q) \le 6\).
'Concave' and 'not concave' describe functions, not sets; the standard term for sets is convex. 'Not convex' is false, since every segment joining two points of an ellipse region stays inside it.
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