A space probe in the shape of the sphere x ^ 2 + y ^ 2 + z ^ 2 = 30 enters Earth's atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (z. y. :) on the probe's surface is T(x, y, z) = x - 2y + 5z Find the hottest point on the probe's surface.
Maximize T(x,y,z)=x-2y+5z
Subject to g(x,y,z)=x2+y2+z2-30=0
"L=T-\\lambda g"
"=x-2y+5z-\\lambda(x^2+y^2+z^2-30)"
"L_x=1-2\\lambda x=0 \\implies2\\lambda x=1\\implies x=\\frac{1}{2\\lambda}"
"L_y=-2-2\\lambda y=0\\implies 2\\lambda y=-2\\implies y=\\frac{-1}{\\lambda}"
"L_z=5-2z\\lambda=0\\implies 2\\lambda z=5\\implies z=\\frac{5}{2\\lambda}"
"g(x,y,z)=(\\frac{1}{2\\lambda})^2+(\\frac{-1}{2\\lambda})^2+(\\frac{5}{2\\lambda})^2-30=0"
"\\frac{1}{4\\lambda^2}+\\frac{1}{\\lambda^2}+\\frac{25}{4\\lambda^2}=30"
"\\frac{1+4+25}{4\\lambda^2}=30"
"\\frac{29}{4\\lambda^2}=30"
"4\\lambda^2*30=29"
"\\lambda^2=\\frac{29}{4*30}" ="\\frac{29}{120}"
"\\lambda=\\sqrt{\\frac{29}{120}}=" 0.4916
"\\lambda=0.4916, x=\\frac{1}{2\\lambda}=1.0171,y=\\frac{-1}{\\lambda}=-2.0342,z=\\frac{5}{2\\lambda}=5.0854"
"T(1.0171,-2.0342,5.0854)=(1.0171-2(-2.0342)+5(5.0854)"
=30.5125
Hottest point on the probe surface is (1.0171,-2.0342,5.0854)
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