Use the Lagrange multiplier method to compute the maximum value of the function $h(x,y,z) = x+z$ on the sphere $x^2+y^2+z^2=1$
My attempt:
∇h = (1,0,1) = λ(2x,2y,2z)
This implies x = z and y = 0
It follows that $2z^2=1$, meaning $z=\pm \sqrt{0.5} = x$
Testing :
$h(\sqrt{0.5}, 0, \sqrt{0.5}) = 2\sqrt{0.5}$
$h(-\sqrt{0.5},0,\sqrt{0.5}) = -2\sqrt{0.5}$
So the maximum value is at $(\sqrt{0.5}, 0, \sqrt{0.5})$