Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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

2 Answers 2

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

share|improve this answer
    
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?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.