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Let f(x,y)=x2+xy+y2f(x, y)=x^2+x y+y^2. (a) Find the maximum and minimum points and values of ff along the circle x2+y2=1x^2+y^2=1. (b) Moving counterclockwise along the circle x2+y2=1x^2+y^2=1, is the function increasing or decreasing at the points (±1,0)(\pm 1,0) and (0,±1)(0, \pm 1) ? (c) Find extreme points and values for ff in the disk DD consisting of all (x,y)(x, y) such that x2+y21x^2+y^2 \leqslant 1.


Answered 1 year ago
Answered 1 year ago
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We have to extremize the function f(x,y)=x2+xy+y2f(x,y)=x^2+xy+y^2 subject to the constraint g(x,y)=x2+y2=1g(x,y)=x^2+y^2=1. Using the method of Lagrange multipliers we have that:

{fx=λgxfy=λgyg(x,y)=1fx=2x+yfy=2y+xgx=2xgy=2y{2x+y=2λxλ=1+y2x2y+x=2λyλ=1+x2yx2+y2=11+y2x=1+x2y    y=±xx2+x2=1x2=12x=±22y=±22\begin{aligned} &\left\{\begin{matrix} f_x=\lambda g_x\\ f_y=\lambda g_y\\ g(x,y)=1 \end{matrix}\right.\\ f_x&=2x+y\\ f_y&=2y+x\\ g_x&=2x\\ g_y&=2y\\ &\left\{\begin{matrix} 2x+y=2\lambda x\to \lambda=1+\dfrac{y}{2x}\\ 2y+x=2\lambda y\to \lambda=1+\dfrac{x}{2y}\\ x^2+y^2=1\\ \end{matrix}\right.\\ 1+\dfrac{y}{2x}&=1+\dfrac{x}{2y}\implies y=\pm x\\ x^2+x^2&=1\\ x^2&=\dfrac{1}{2}\\ x&=\pm\dfrac{\sqrt{2}}{2}\\ y&=\pm\dfrac{\sqrt{2}}{2} \end{aligned}

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