# Limits/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

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

• Find the critical points:

A graph shows:

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:

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Wow! Nice work! +1 –  amWhy Sep 1 '13 at 14:07
@amWhy: Did you see this idea? I love the idea of having professors interacting with the younger students. proofschool.org/#1 –  Amzoti Sep 1 '13 at 17:24
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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