Discussion
Let \(f, g : \mathbb{R}^{2} \to \mathbb{R}\) be defined by
\[ f(x,y) = x^{2} - \frac{3}{2}xy^{2} \]
and
\[ g(x,y) = 4x^{4} - 5x^{2}y + y^{2} \]
for all \((x,y) \in \mathbb{R}^{2}\).
Consider the following statements:
P: f has a saddle point at (0,0).
Q: g has a saddle point at (0,0).
Then
Answer: A
Both P and Q are TRUE.
Both functions have zero gradient at the origin, and their Hessians there are only semi-definite, so the second derivative test is inconclusive; look at the sign of each function near (0, 0).
f: on the x-axis \(f = x^{2} \gt 0\). On the curve \(x = y^{2}\), \(f = y^{4} - \tfrac{3}{2}y^{4} = -\tfrac{1}{2}y^{4} \lt 0\). So f takes both signs in every neighbourhood of the origin, where f = 0: a saddle point.
g factorises:
\[ g = (y - x^{2})(y - 4x^{2}) \]
It is negative between the parabolas \(y = x^{2}\) and \(y = 4x^{2}\) (for example at \(y = 2x^{2}\), \(g = -2x^{4}\)) and positive elsewhere near the origin, such as on the x-axis. So g also has a saddle point, although it has a minimum along every straight line through the origin.
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