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Section 3 Key Concepts

  • \((a,b) \) is a local maximum of \(f(x,y) \) if \(f(a,b)\geq f(x,y) \) for all \((x,y) \) near \((a,b) \text{.}\) Similarly, \((a,b) \) is a local minimum of \(f(x,y) \) if \(f(a,b)\leq f(x,y) \) for all \((x,y) \) near \((a,b) \text{.}\)
  • A critical point of a function \(f(x,y) \) is a point where \(\nabla f(x,y)=\vec{0} \) or where \(\nabla f(x,y) \) is undefined.
  • Any local minima or maxima of \(f(x,y) \) must occur at a critical point of \(f(x,y) \text{.}\)

  • The nature of a critical point can be determined using the Second Derivative Test. Specifically, let

    \begin{equation*} H_f=\left[ \begin{array}{cc} f_{xx} \amp f_{xy} \\ f_{yx} \amp f_{yy} \end{array}\right] \end{equation*}

    and let \(D=\det{(H_f)} \) (\(H_f \) is called the Hessian matrix of \(f(x,y) \)). If \((a,b) \) is a critical point of \(f(x,y) \text{,}\) then:

    1. if \(D(a,b)\gt 0 \) and \(f_{xx}\gt 0 \text{,}\) \((a,b) \) is a local minimum
    2. if \(D(a,b)>0 \) and \(f_{xx}\lt 0 \text{,}\) \((a,b) \) is a local maximum
    3. if \(D(a,b)\lt 0 \text{,}\) then \((a,b) \) is a saddle point (neither a maximum or a minimum)
    4. if \(D(a,b)=0 \text{,}\) the test is inconclusive.

    Note: If the test is inconclusive or \(\nabla f \) is undefined, we need to use some other method.

  • Other methods to determine the nature of a critical point of \(f(x,y) \) include looking at its graph or a contour diagram.
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