Math_Struggles

What must Ax equal for the equations f(x) = (Ax \cdot x)e^{||x||^{2}} to have a critical point? How would I prove that. Or at least start a proof? Thanks!

I need to find what Ax must equal in order to for (f(x) = (Ax(dot)x)(e^(||x||^2))) to have a critical point. I've gotten it down to Ax = -x(Ax dot x) but I dont think that's right. I'm really not sure.

Yes but we don't know that B is necessarily a diagonal matrix. Does it matter?

How would I prove that if x is a critical point, Ax = (lambda)Bx?

Yeah I used the quadratic equation and got [math] y = (-1+\sqrt{2})x [/math] [math] y = -(1+\sqrt{2})x [/math] Now I don't know how to use that to find the critical points. Is it any x and y that fulfill that? And if it's any, meaning there are infinite critical points, how do I know which is the max or min?

I need to find the critical points before I can use the second derivative test. The problem is, instead of getting a number for x and y, I get expressions of one in terms of the other, and I don't know what to do with that.

I thought I said that we are trying to prove that Q(x) has at least one critical point, where Q(x) is the Generalize Rayleigh Quotient (Ax dot x)/(Bx dot x). And it is an undergraduate homework problem, it's just the weekend so the teacher doesn't have office hours... The homework is for a multivariable calculus class. This is only the first part of the problem too. The given information is that A and B are real symmetric n x n matrices, and B is positive definite. (a) I first need to prove Q has at least one critical point. (b) Then I need to prove that x is a CP of Q iff Ax = (lambda)Bx where lambda is a real eigenvalue. © Next I need to prove that lambda from b satisfies lambda = Q(x). (D) Finally, I need to prove that in part, det(A-(lambda)B) = 0. I have no idea where to start, our teacher has a habit of giving us extremely difficult homework problems, but she usually has office hours so we may get help. Thank you in advance for any help you can give me!

Yes that is what I did. I took the partial derivatives with respect to x and y and thensimplified them down to that result. I'll attemt to show my work here, but I don't know how to make the work look like real math steps rather than typed out math. f(x) = partial derivative with respect to x f(y) = same with y f(X) = 0 = 2((x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y))/(x^2-2xy+3y^2)^2 f(x) = 0 = (3x^3-6x^2y+9xy^2+x^2y-2xy^2+3y^3-3x^3-2x^2y-xy^2+3x^2y+2xy^2+y^3)/(x^2-2xy+3y^2)^2 f(x) = 0 = (-4x^2y+8xy^2+4y^3)/(x^2-2xy+3y^2)^2 f(x) = 0 = (y^3+2xy^2-x^2y)/(x^2-2xy+3y^2)^2 The only way that f(x) will be zero is if the numerator is zero, so we set it equal to zero: 0 = y^3+2xy^2-x^2y 0 = y(y^2+2xy-x^2) I realize now that when I wrote y(x-y)^2 I was wrong in that y^2+2xy-x^2 does not factor to (x-y)^2! f(y) = 0 = 2((x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y))/(x^2-2xy+3y^2)^2 The work is very similar for f(y), and I end up with 0 = x(x^2-2xy-y^2) So now I don't know where to go from there. I see that the f(x) equation can be rewritten as 0 = y(x^2-2xy-y^2), but I don't know how that really helps matters. Thanks!

I've simplified it to the equations 0 = x(x-y)^2 and 0 = y(x-y)^2. I know that x and y can't be zero because at (0,0) the function is undefined. So are the critical points anywhere where x = y? That seems to be the place that the other quantity in the two equations would be zero.

Can someone please help me find the minimum and maximum of the following equation: (3x^2 + 2xy + y^2)/(x^2 - 2xy + 3y^2) Please show work! Thanks!

We only know that B is positive definite. I have no idea where to start. If I had some idea where to start I think I could figure much of it out, but I really have on clue! Any help is so much appreciated. Thanks!

I forgot some parentheses: It should be (x^tAx)Bx = (x^tBx)Ax If that's not right can you please help with proving that the Generalized Rayleigh Quotient has at least one critical point?

I really need some help with this proof: If A and B are nxn symmetric matrices, and x is a column vector with x^t as it's transpose: Is it true that x^tAxBx = x^tBxAx? I arrived at this because I am trying to prove that the Generalized Rayleigh Quotient (Ax dot x)/(Bx dot x) has at least one critical point. If the first statement is not true, please help me with the second part. Thanks! There will be more proofs to follow after this if I run into more trouble, because the problem is multiple steps of proofs.