Given that [tex]x+y=1[/tex], we have [tex]y=1-x[/tex], so that
[tex]f(x,y)=x^2+y^2\iff g(x)=x^2+(1-x)^2[/tex]
Take the derivative and find the critical points of [tex]g[/tex]:
[tex]g'(x)=2x-2(1-x)=4x-2=0\implies x=\dfrac12[/tex]
Take the second derivative and evaluate it at the critical point:
[tex]g''(x)=4>0[/tex]
Since [tex]g''[/tex] is positive for all [tex]x[/tex], the critical point is a minimum.
At the critical point, we get the minimum value [tex]g\left(\frac12\right)=f\left(\frac12,\frac12\right)=\frac12[/tex].
