In one line
At a stationary point compute \(r=f_{xx}\), \(s=f_{xy}\), \(t=f_{yy}\): if \(rt-s^2>0\) it is a maximum when \(r<0\) and a minimum when \(r>0\); if \(rt-s^2<0\) it is a saddle point.
Explained simply
In one variable you check \(f''\): negative means the curve bends down (hill top), positive means it bends up (valley). In two variables the surface can bend differently in different directions, so one number is not enough. The quantity \(rt-s^2\) checks whether the bending is the same in every direction. If it is (\(rt-s^2>0\)), the sign of \(r\) tells you up or down. If not (\(rt-s^2<0\)), the surface goes up one way and down another: a saddle.
This is Lagrange's condition for extreme values. The case \(rt-s^2=0\) is doubtful and needs further investigation.
How it works
- Find all stationary points from \(f_x=0\), \(f_y=0\).
- Find \(r=f_{xx}\), \(s=f_{xy}\), \(t=f_{yy}\) as functions of \(x,y\).
- At each point, evaluate \(r,s,t\) and \(rt-s^2\).
- Classify using the table below and compute \(f\) at maxima and minima.
| \(rt-s^2\) | \(r\) | Nature |
|---|---|---|
| \(>0\) | \(<0\) | Maximum |
| \(>0\) | \(>0\) | Minimum |
| \(<0\) | any | Saddle point (neither) |
| \(=0\) | any | Doubtful, test further |
Key formula
\(r=\dfrac{\partial^2f}{\partial x^2},\ s=\dfrac{\partial^2f}{\partial x\partial y},\ t=\dfrac{\partial^2f}{\partial y^2}\), all evaluated at the stationary point.
Quick example
\(f=x^3+y^3-3axy\), \(a>0\).
\(f_x=3x^2-3ay=0\), \(f_y=3y^2-3ax=0\). From the first \(y=x^2/a\); substituting, \(x^4/a^2=ax\Rightarrow x(x^3-a^3)=0\). Points: \((0,0)\) and \((a,a)\).
\(r=6x\), \(s=-3a\), \(t=6y\).
- At \((a,a)\): \(rt-s^2=36a^2-9a^2=27a^2>0\), \(r=6a>0\): minimum. Value \(a^3+a^3-3a^3=-a^3\).
- At \((0,0)\): \(rt-s^2=0-9a^2<0\): saddle point.
Common mistakes
- Using the sign of \(r\) when \(rt-s^2<0\); then it is a saddle regardless of \(r\).
- Writing \(rt+s^2\) or \(r t-s\) by mistake.
- Evaluating \(r,s,t\) at a general point instead of at each stationary point.
How MU asks it
Asked as 5/6-mark parts: "Discuss the maxima and minima of \(f(x,y)=x^3y^2(1-x-y)\)" or "Find the extreme values of \(x^2+y^2+6x+12\)". Present stationary points and the \(r,s,t,rt-s^2\) values in a small table; examiners like that. State the nature and the value clearly for each point.