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alexandr1967 [171]
3 years ago
10

Use lagrange multipliers to find the points on the given surface that are closest to the origin. y2 = 64 + xz

Mathematics
1 answer:
Inessa [10]3 years ago
8 0
The distance between an arbitrary point on the surface and the origin is

d(x,y,z)=\sqrt{x^2+y^2+z^2}

Recall that for differentiable functions g(x) and h(x), the composition g(h(x)) attains extrema at the same points that h(x) does, so we can consider an augmented distance function

D(x,y,z)=x^2+y^2+z^2

The Lagrangian would then be

L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(y^2-64-xz)

We have partial derivatives

\begin{cases}L_x=2x-\lambda z\\L_y=2y+2y\lambda\\L_z=2z-\lambda x\\L_\lambda=y^2-64-xz\end{cases}

Set each partial derivative to 0 and solve the system to find the critical points.

From the second equation it follows that either y=0 or \lambda=-1. In the first case we arrive at a contradiction (I'll leave establishing that to you). If \lambda=-1, then we have

\begin{cases}2x+z=0\\2z+x=0\end{cases}\implies x=0,z=0

This means y^2=64\implies y=\pm8

so that the points on the surface closest to the origin are (0,\pm8,0).
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