.
with
z =(4/3) -(x/3)-(2y/3). That is, the function
.
Computing partials:
.
.
(Typo in last line above:last line: should have 2 x + 13 y = 8. The correct equation is used below.)
Next,
Finally, z =(4/3) -(x/3)-(2y/3) = 4/3 -2/21 -8/21 = 6/7 .
Here is a second way to solve this problem, without calculus:
The plane x + 2y + 3z = 4 has the normal <1, 2, 3>.
The line <1, 2, 3> t is the equation of the
normal through the origin. This line intersects the
plane when t + 2(2t) + 3(3t) = 4.
That is, when
14t = 4 or t = 2/7, or (x, y, z) = (2/7, 4/7, 6/7). The length of
this normal will be the minimal distance
from the plane to the origin.