AcademySeveral-Variable Calculus
Academy
Critical Points
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
Critical points of multivariable functions are candidates for local maxima, local minima, or saddle points. They occur where the gradient is zero or undefined.
Notation
\(\nabla f\)
gradient of f
\(f_{xx},f_{yy},f_{xy}\)
second partial derivatives
\(D\)
second derivative test determinant
The Core Method
For differentiable \(f(x,y)\), solve
Critical point condition
\[\nabla f=(0,0)\]
For the second derivative test, compute
Discriminant
\[D=f_{xx}f_{yy}-(f_{xy})^2\]
If \(D>0\) and \(f_{xx}>0\), there is a local minimum. If \(D>0\) and \(f_{xx}<0\), there is a local maximum. If \(D<0\), there is a saddle point.
Worked Cases
Question
Classify the critical point of \(f(x,y)=x^2+y^2\).
Answer
The gradient is \(\nabla f=(2x,2y)\), so the only critical point is \((0,0)\). The second partials are \(f_{xx}=2\), \(f_{yy}=2\), and \(f_{xy}=0\). Thus \(D=2\cdot2-0=4>0\), and \(f_{xx}>0\). Therefore \((0,0)\) is a local minimum.
Examples
Question
Classify \((0,0)\) for \(f(x,y)=x^2-y^2\).
Answer
Here \(f_{xx}=2\), \(f_{yy}=-2\), and \(f_{xy}=0\). Then \(D=2(-2)-0=-4<0\). Therefore \((0,0)\) is a saddle point.