A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

zmudz

  • one year ago

For \(x\), \(y\), and \(z\) positive real numbers, what is the maximum possible value for \( \sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+5y+4z}}? \)

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do you know how to solve it

  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Apparently you're dealing with multivariable Calculus, you have a function $$f: \mathbb{R}^3 \to \mathbb{R}$$ which is called a real valued multivariable function (real valued because it's codomain are the reals, multivariable because it takes more than one value as arguments). Similar as in real analysis to obtain a Maximum/Minimum you must figure out the critical points first. In real Analysis this is done by checking for which \( x \in \mathbb{R}\) the equation \( f(x)=0\) is satisfied. In your case you do something very similar, just that the derivative isn't as straight forward anymore. You want to check for the Gradient of f to be annihilated, that is \( \nabla f(x)=0 \in \mathbb{R}^3 \). So take all the partial derivatives of $f$ (with respect to x,y,z) and put them into your vector and evaluated the equation above. This will give you the critical points, in order to be sure that it is a maximum you need to do again something very similar than in real analysis, rather than checking for the sign of \(f''(x)\) you need to discuss the Hessian Matrix of \(f\) which is the symmetric Matrix (because your function is \(C^2\) semi smooth) with entries evaluated at the critical points. If said Matrix is positive definite, you have a (local) minimum, if it is positive definite you have a (local) maximum. Make sure that you check for the definitions of all the words above that I have said in order to solve your problem, that is: - Gradient of a function - Hessian Matrix of a function - Definiteness of a Matrix

  3. zmudz
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    This is for a precalc class though :/

  4. ChillOut
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I can't see any way to solve this withotu calculus;

  5. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    All of these are based on inequalities. Review your inequalities between harmonic, geometric, arithmetic and quadratic means, and whethever generalization you covered in class.

  6. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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
  • 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.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.