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Section 6 Wrap Up Questions

  1. Answer the following as True or False with a brief reason.

    1. If the contours of \(f(x,y)\) get closer together as you move in the \(y\)-direction from point \((a,b)\text{,}\) then \(f_{yy}(a,b)>0\text{.}\)

    2. There are \(8\) second partial derivatives of \(f(x,y,z)\text{.}\)

    3. There is no function \(f(x,y)\) for which \(f_{xx}=f_{yy}\text{.}\)

  2. We have two different notations for the mixed partial derivative with respect to \(x\) and then \(y\text{:}\) \(f_{xy}\) and \(\displaystyle \frac{\partial^2 z}{\partial y \partial x}\text{.}\) Explain why the order of \(x\) and \(y\) appears to be inconsistent in the two notations.

  3. Describe the basic shape of a contour diagram for a function \(f(x,y)\) near a point \((a,b)\) if you know that \(f_{x}(a,b)>0\text{,}\) \(f_{y}(a,b)\lt 0\text{,}\) \(f_{xx}(a,b)\lt 0\text{,}\) \(f_{yy}(a,b)\lt 0\text{.}\)

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