Show that the distance D from the origin of any point (x , y , z) lying on the plane x + y + z= 1 satisfies
D^2 + x^2 + y^2 +(x + y -1)^2 .
By considering partial derivatives, find the point that is closest to the origin. Prove that this distance is genuinely minimal.
The Attempt at a Solution
I did the first part. Knowing Distance is D^2 = x^2 + y^2 + z^2 then by re-arranging x + y + z =1 to -z = x + y - 1.
z^2 = (-z)^2 = (x + y - 1)^2
D^2 = x^2 + y^2 + (x + y - 1)^2
Not sure how to do the next part. How does considering partial derivatives help me find a point closest to the origin?
Any help would be great.