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Question: example 3 the temperature of a point x y z...

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Example 3: The temperature of a point (x, y, z) on the unit sphere is given by T(x, y,z) xy + yz. Find the temperature of the hottest point on the sphere. Solution: Maximise f- xy + yz subject to 5-x2 + y2 + z2-1 using the Lagrange multiplier method. g-1 0 Writing out the components:
If we assume x, y, z f 0, we can rewrite: From which we obtain that x = z and 2x2-y2 Substituting into the constraint g = x2 + y2 + z2 1 gives So we have the following stationary points t(1/2,1//2, 1/2) and (1/2,-1//2, 1/2) with values: T(1/2,1/v2,1/2)1/2 and T(-1/2,-1/2,-1/2)12
Now consider x 0, From the equations (f-λ8) 0 we find that y z 0 which does not satisfy the constraint g1. Similarly taking z 0 gives x0 which does not satisfy the constraint g-1 However we can find a solution to V(f - g)0 with y 0 with non-zero x & z which gives additional stationary points: Values at these points are: V2, 0, 1 The temperature is maximum at +(1/2,1/V2,1/2)
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