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3d minimina/maxima problem

  1. May 21, 2009 #1
    1. The problem statement, all variables and given/known data

    3. The height of a certain hill (in feet) is given by
    h(x, y) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12)
    where y is the distance (in miles) north, x is the distance (in miles)
    east of the village.
    (a) Where is the top of the hill located?
    (b) How high is the hill?
    (c) How steep is the slope (in feet per mile) at a point 1 mile north
    and one mile east of the village? In what direction is the slope
    steepest at that point?

    2. Relevant equations

    3. The attempt at a solution

    I would think that I would have to use the derivative test to get the extrema and go from there, but we didn't cover that material yet. How else can I approach this?
  2. jcsd
  3. May 21, 2009 #2
    Just take the partial derivatives of h, set them both equal to zero, and solve the system. Then test the points to see which one gives you the largest value for h.

    Find the gradient for part (c).
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