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Please help me with this question. It's topic on limits and partial derivative.

Find the maximal value of $f(x,y)=xy((1-x^2/a^2-y^2/b^2))^{0.5}$ for $a=74.4$, $b=64.8$. Round off your answer to $4$ decimal places.

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What have you done so far? Is the exponent $ 0.5$ for the last term in the objective fn? – Macavity Sep 1 '13 at 11:01
up vote 2 down vote accepted

I am going to provide hints and you can fill in the details.

  • Find the critical points:

A graph shows:

enter image description here

The critical points are located at:

$(0,0), (-42.95486002770815687948066927, -37.41229744348774954019284098), (42.95486002770815687948066927, 37.41229744348774954019284098), (-42.95486002770815687948066927, 37.41229744348774954019284098), (42.95486002770815687948066927, -37.41229744348774954019284098)$

  • Next, find all partials: $f_x, f_y, f_{xx}, f_{yy}, f_{xy}, f_{yx}$
  • Using those partials and if possible, classify each of these critical points as a global or local minimum or maximum
  • You should arrive at a global max $927.824976598496188596782456$ at two of the points
  • You should arrive at a global min $-927.824976598496188596782456$ at two of the points
  • The other critical point cannot be classified

As an additional hint, here is a 3D plot of this function:

enter image description here

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Wow! Nice work! +1 – amWhy Sep 1 '13 at 14:07
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Awesome! Another "favorited" website! Thanks ;-) – amWhy Sep 1 '13 at 17:28

Just a comment, but I find it easier to enter math as an answer:

It seems to me that it would be easier to work with $f^2(x,y)=x^2y^2(1-\frac{x^2}{a^2}-\frac{y^2}{b^2}) $.

Also, is the fact that this is dealing with the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} =1$ of any help?

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