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Find the maximum value of $Q(\mathbf{x})=-3 x_{1}^{2}+5 x_{2}^{2}-2 x_{1} x_{2}$ subject to the constraint $x_{1}^{2}+x_{2}^{2}=1 .$ (Do not go on to find a vector where the maximum is attained.)

$1+\sqrt{17}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 3

Constrained Optimization

Introduction to Matrices

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Okay, So in this problem, we need to find the Mexican value off acute acts without, um without finding the vector where the Mexican he's attained. So this is actually we need to apply our, um, our knowledge and Nash bro. So at first, the first, the first thing we need to remark here is the inequality X squared plus x two square speaker or equal to two excellent x two. So that means our to X one x two will be smaller or equal to one. And so our ex likes to will be smaller or equal to 1/2. Now recall her or assumption of q X, which is connective three x one square. That's five x two squared minus two x one x two So minus two x y excuse. We just apply this inequality times two times Negative, too. 22 Ex LAX, too. It's bigger. We could to one negative one. So what that means this expression will be bigger. We put too negative. Three x one squared plus five excuse Weird. Minus one. Okay. And now we can also take X one squared to be equal to one minus x two square. It's a spiral assumption. So we have negative three one minus x two squared 1st 5 X two squared minus one. So that is active three US three excuse squared US five x two squared my one. So that is aid X two squared minus four. So no, still to make the to make you accept as large as possible So we can check that the lower bound off co ax, which is eight x two squared minus four So we can't check how large this lower bound can be. So that means we need to take X two s started. It's possible the biggest X two is one and x two cannot be feared or what? Because if that extreme is bigger than one than our our assumption next one squared plus X two squared, it was one will be filled. So the largest X do we can take just one. So at this moment, X one will be easy so that I hear the expression here will be equal to, uh, four. Now, we just plugging the set of values are facts into our, um we're ready for So we have Q um, que axe Hope? Sorry. Q. Max is equal to now x Y zero. So this term will be zero X two is one. So here will be five and X Y zero. So these two will be zero. So we have fine off course 55 speaker or two for so that is a Mexican valley we can attain.

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