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Section 7 Wrap Up Questions

  1. Are the following statements true or false?

    1. The only places a local minima or maxima can occur are where \(\nabla f=\vec{0} \)
    2. Provided \(\nabla f\neq \vec{0} \text{,}\) the second derivative test can always be used to determine the nature of a critical point
  2. Explain geometrically why, if \(f_x \) and \(f_y \) are defined, they both have to be zero at a minimum or maximum.
  3. What do you expect a contour diagram to look like close to a minimum or maximum? What about close to a saddle point?
  4. If two contours cross at a point \((a,b) \text{,}\) does this mean it has to be a critical point? If so, does it have to be a saddle?
  5. What do we do if \(D=0\text{?}\) What are reasonable strategies?
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