Mathematics: Find the global maximum and global minimum values (if they exist) of x 2 + y 2 in the region x + y = 1. If there is no global max, justify why. If there is no global minimum, justify why.

The slope is 2 and -7 is the y intercept...

Find the global maximum and global minimum values

lindsay45

20.01.2022

Given that x+y=1 , we have y=1-x , so that

$f(x,y)=x^2+y^2\iff g(x)=x^2+(1-x)^2$

Take the derivative and find the critical points of :

$g'(x)=2x-2(1-x)=4x-2=0\implies x=\dfrac12$

Take the second derivative and evaluate it at the critical point:

g''(x)=40

Since g'' is positive for all , the critical point is a minimum.

At the critical point, we get the minimum value $g\left(\frac12\right)=f\left(\frac12,\frac12\right)=\frac12$ .

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