Discussion
Let \(f: \mathbb{R}^{2} \to \mathbb{R}\) be the function given by
\(f(x,y) = (x^{2} - 1)^{2} + (y^{2} - 1)^{2}\) for all \((x,y) \in \mathbb{R}^{2}\).
Which ONE of the following statements is TRUE?
Answer: D
The TRUE statement is: \(f\) has exactly four saddle points.
Write \(f(x,y) = g(x) + g(y)\) with \(g(t) = (t^{2} - 1)^{2}\). Since \(g'(t) = 4t(t^{2} - 1)\), \(g\) has local minima at \(t = \pm 1\) (value 0) and a local maximum at \(t = 0\) (value 1). The critical points of \(f\) are the nine points with \(x, y \in \{-1, 0, 1\}\), and their type follows from the two coordinates:
- \((\pm 1, \pm 1)\): a minimum in both directions, so 4 local minima, where \(f = 0\).
- \((0,0)\): a maximum in both directions, so 1 local maximum, where \(f = 2\).
- \((0, \pm 1)\) and \((\pm 1, 0)\): a maximum in one direction and a minimum in the other, so 4 saddle points.
So there is only one local maximum, not two, and there are four local minima, not two or three.
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