A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 5 years ago
Use Lagrange multipliers to find the maximum or minimum values of f(x,y) subject to the constraint.
f(x,y)= x^2+y, x^2y^2=1
anonymous
 5 years ago
Use Lagrange multipliers to find the maximum or minimum values of f(x,y) subject to the constraint. f(x,y)= x^2+y, x^2y^2=1

This Question is Closed

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0We have the function \[f(x,y)=x^2+y\] and the constraint \[g(x,y)=x^2y^2=1\] which is a hyperbola with the y=x and y=x lines as asymptotes. Let's introduce the auxiliary function \[\Lambda(x,y,\lambda)=f(x,y)\lambda(g(x,y)1).\] Finding the critical points of this function is equivalent to finding the critical points of f(x,y) subject to the constraint. Compute the partial derivatives: \[\frac{\partial\Lambda}{\partial x},\quad\frac{\partial\Lambda}{\partial y},\quad\frac{\partial\Lambda}{\partial \lambda}\] then solve the system of linear equations \[\begin{array}{rcl}\frac{\partial\Lambda}{\partial x}&=&0,\\\frac{\partial\Lambda}{\partial y}&=&0,\\\frac{\partial\Lambda}{\partial\lambda}&=&0.\end{array}\] You'll get the solution \[x=\pm\frac{\sqrt{5}}{2},\quad y=\frac{1}{2},\quad\lambda=1.\] (You have to use that x cannot be zero on the hyperbola.) If you compute the value of f at the points \[(\frac{\sqrt{5}}{2},\frac{1}{2})\textrm{ and }(\frac{\sqrt{5}}{2},\frac{1}{2})\] then you'll get \[f(\frac{\sqrt{5}}{2},\frac{1}{2})=\frac{3}{4}\textrm{ and }f(\frac{\sqrt{5}}{2},\frac{1}{2})=\frac{3}{4}.\] You could use the Hessian matrix of Lambda the find out whether these point are maximum or minimum. They turn out to be minimums. Solution 2: A much easier way to find these points is use that\[x^2=1+y^2\textrm{ (this is the constraint)}\] and substitute the righthand side to f. Then \[f(x(y),y)=1+y^2+y\] is just a single variable function (namely a parabola) which has a minimum at point y=1/2. From the constraint the values of x will be\[x=\pm\frac{\sqrt{5}}{2}.\]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.