Maximize $x_1^2 + 4x_1 x_2 +x_2^2$ subject to $x_1^2 +x_2^2 =1$

Question

[If someone could please show me how to complete this problem I would be really grateful, the instructor didn't provide an answer key for the old practice exams.

Answer 1

As , hints given by $\text{Toby Mak} $ ,

If you take $x_1=cosx , x_2= sinx $

Then , as, $x_{1}^{2}+x_{2}^{2}+4x_{1}x_{2} = cos^{2}x+sin^{2}x+4cosx\cdot sinx = 1 +4cosx\cdot sinx $

So, we have to find maximum value of $cosx\cdot sinx $

Take , $f(x)=cosx\cdot sinx $

Now , $f'(x)=cos^{2}x-sin^{2}x = cos(2x) $

Maximum will be taken by $f$ at $f'(x)=0 \implies cos(2x)=0 \implies x=\frac{\pi}{4} $

Now , at $x=\frac{\pi}{4} $ , $f=\frac{1}{2} $

So, Max of $x_{1}^{2}+x_{2}^{2}+4x_{1}x_{2}=1+\frac{4}{2}=3 $