James Stewart

ISBN: 9781305266643

Textbook Problem

Find the points on the surface y² = 9 + xz that are closest to the origin.

To determine

To find: The points on the surface y2=9+xz which is closest to the origin.

The surface equation is, y2=9+xz .

Calculation:

The distance d between (0,0,0) and any point (x,y,z) is,

d=(x−0)2+(y−0)2+(z−0)2d2=x2+y2+z2

Substitute the value y2=9+xz in the above distance equation.

Thus, d2=x2+9+xz+z2

Let, f(x,y)=x2+9+xz+z2 .

Take the partial derivative of f(x,y) with respect x and obtain fx .

∂f∂x=∂∂x(x2+9+xz+z2)=∂∂x(x2)+∂∂x(9)+∂∂x(xz)+∂∂x(z2)=2x+0+z(1)+0=2x+z

Hence, fx=2x+z (1)

Take the partial derivative of f(x,y) with respect z and obtain fz

Check out a sample textbook solution.

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Calculus Multivariable Calculus 8th Edition

Find the points on the surface y² = 9 + xz that are closest to the origin.