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Second Derivative Test for Partial Derivatives

  1. Dec 19, 2011 #1
    Hi there, just wanted to make a clarification before my final exam.

    The second derivative test for partial derivatives (or at least part of it) states

    if D = ∂2f/∂x2 * ∂2f/∂y2 - (∂2f/∂x∂y)2 and (a,b) is a critical point of f, then

    a) if D(a,b) > 0 and ∂2f/∂x2 < 0, then there is a local max at (a,b)

    b)if D(a,b) > 0 and ∂2f/∂x2 > 0, then there is a local min at (a,b)

    and the other two parts are irrelevant for my question. My question is, do I have to specifically check that ∂2f/∂x2 is positive or negative, or can I check that ∂2f/∂y2 is positive or negative instead? i.e. does it really matter which one I check? Thank you in advance
     
  2. jcsd
  3. Dec 20, 2011 #2

    HallsofIvy

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    Clearly, [itex]\left(\partial^2 f/\partial x\partial y\right)^2[/itex] is positive and you are subtracting it from [itex]\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)[/itex]. In order that the difference be positive, [itex]\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)[/itex] must be positive which means that [itex]\partial^2f/\partial x^2[/itex] and [itex]\partial^2 f/\partial y^2[/itex] must have the same sign.
     
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