## zmudz one year ago For $$x$$, $$y$$, and $$z$$ positive real numbers, what is the maximum possible value for $$\sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+5y+4z}}?$$

1. anonymous

do you know how to solve it

2. anonymous

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

This is for a precalc class though :/

4. ChillOut

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

5. beginnersmind

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