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Question

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

Solution

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### (a)

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

\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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