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jolli1 [7]
3 years ago
15

Consider the function f(x, y) = x2 + xy + y2 defined on the unit disc, namely, d = {(x, y)| x2 + y2 ≤ 1}. use the method of lagr

ange multipliers to locate the maximum and minimum points for f on the unit circle. use this to determine the absolute maximum and minimum values for f on
d. maximum minimum
Mathematics
1 answer:
Pani-rosa [81]3 years ago
8 0
First we note that the partial derivatives vanish simultaneously at one point:

\begin{cases}f_x=2x+y=0\\f_y=x+2y=0\end{cases}\implies(x,y)=(0,0)

so there is one critical point within the region D.

The Lagrangian is

L(x,y,\lambda)=x^2+xy+y^2+\lambda(x^2+y^2-1)

and has partial derivatives

L_x=2x+y+2\lambda x
L_y=x+2y+2\lambda y
L_\lambda=x^2+y^2-1

Set each partial derivative to 0 to find the possible critical points within the disk D. Then we notice that

yL_x=2xy+y^2+2\lambda xy=0
xL_y=x^2+2xy+2\lambda xy=0
L_\lambda=0\implies x^2+y^2=1

\implies xL_y-yL_x=x^2-y^2=0

Since x^2+y^2=1, we have

x^2-y^2=x^2+y^2-2y^2=0\implies1=2y^2\implies y^2=\dfrac12\implies y=\pm\dfrac1{\sqrt2}

And since x^2-y^2=0, or x^2=y^2, we also have

x=\pm\dfrac1{\sqrt2}

So we have four possible additional critical points to consider:

f(0,0)=0
f\left(-\dfrac1{\sqrt2},-\dfrac1{\sqrt2}\right)=\dfrac32
f\left(-\dfrac1{\sqrt2},\dfrac1{\sqrt2}\right)=\dfrac12
f\left(\dfrac1{\sqrt2},-\dfrac1{\sqrt2}\right)=\dfrac12
f\left(\dfrac1{\sqrt2},\dfrac1{\sqrt2}\right)=\dfrac32

It should be clear enough which of these correspond to the absolute extrema of f over D.
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