In one line
A saddle point is a stationary point that is a maximum along one direction and a minimum along another, so it is neither a maximum nor a minimum; it shows up as \(rt-s^2<0\).
Explained simply
Look at a horse saddle. Along the horse's spine the saddle curves up at the front and back, so the middle is a low point. Across the horse, from one side to the other, it curves down, so the middle is a high point. At the centre the surface is flat (stationary), but it is not a top or a bottom. That centre is a saddle point. A mountain pass between two peaks is another example.
Technically, \((a,b)\) is a saddle point of \(f\) if \(f_x=f_y=0\) there and every neighbourhood contains points where \(f>f(a,b)\) and points where \(f<f(a,b)\). The second derivative test detects it: \(rt-s^2<0\).
How it works
- Find stationary points.
- Compute \(rt-s^2\) at each one.
- If it is negative, call the point a saddle point; there is no maximum or minimum value there.
- Optional check: move along two different lines through the point and see \(f\) rise along one and fall along the other.
Key formula
Saddle point condition: \(f_x=f_y=0\) and \(rt-s^2<0\), where \(r=f_{xx}\), \(s=f_{xy}\), \(t=f_{yy}\).
Quick example
\(f(x,y)=x^2-y^2\).
\(f_x=2x=0\), \(f_y=-2y=0\): stationary point \((0,0)\), \(f(0,0)=0\).
\(r=2\), \(s=0\), \(t=-2\), so \(rt-s^2=-4<0\): saddle point.
Check: along the \(x\)-axis \(f=x^2\ge0\) (minimum at the origin), along the \(y\)-axis \(f=-y^2\le0\) (maximum at the origin). So the origin is neither.
Another: \(f=xy\) has \(r=0,s=1,t=0\), \(rt-s^2=-1<0\); the origin is a saddle point (positive in the first quadrant, negative in the second).
Common mistakes
- Calling a saddle point a "point of inflexion"; for two variables the correct term is saddle point.
- Concluding "minimum" because \(r>0\), ignoring that \(rt-s^2<0\).
- Thinking \(rt-s^2=0\) means saddle; it means the test fails and more work is needed.
How MU asks it
Saddle points are not asked as a separate question; they appear inside 5/6-mark maxima–minima questions, where one of the stationary points is usually a saddle. Write "\(rt-s^2<0\), hence \((a,b)\) is a saddle point; \(f\) has neither maximum nor minimum there." Missing this line costs a mark.