Discussion
Consider the function
\[ f(x, y) = x^{2}y + 2xy^{2} - 2x^{2}y^{2}. \]
Then which one of the following statements is correct ?
Answer: D
\((\frac{3}{2}, \frac{3}{4})\) is a saddle point of f.
At \((\frac{3}{2}, \frac{3}{4})\) both first derivatives vanish:
\[ \begin{aligned} f_{x} &= 2xy+2y^{2}-4xy^{2} = 2.25+1.125-3.375 = 0 \\ f_{y} &= x^{2}+4xy-4x^{2}y = 2.25+4.5-6.75 = 0 \end{aligned} \]
The second derivatives there are:
\[ \begin{aligned} f_{xx} &= 2y-4y^{2} = -0.75 \\ f_{yy} &= 4x-4x^{2} = -3 \\ f_{xy} &= 2x+4y-8xy = -3 \end{aligned} \]
\(f_{xx}f_{yy}-f_{xy}^{2} = 2.25 - 9 \lt 0\), so the point is a saddle, not a maximum. At \((\frac{3}{2}, 0)\) and \((0, \frac{3}{4})\) the gradient is not zero (\(f_{y} = 2.25\) and \(f_{x} = 1.125\) respectively), so they are not extreme points at all.
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