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Ede4ka [16]
3 years ago
10

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 ma

x, justify why. If there is no global minimum, justify why.
Mathematics
1 answer:
Grace [21]3 years ago
5 0

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:

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)=4>0

Since g'' is positive for all x, 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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