   Chapter 14, Problem 63RE

Chapter
Section
Textbook Problem

Find the points on the surface xy2z3= 2 that are closest to the origin.

To determine

To find: The points on the surface xy2z3=2 which is closest to the origin.

Explanation

Given:

The surface equation is, xy2z3=2.

Calculation:

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

d=(x0)2+(y0)2+(z0)2d2=x2+y2+z2

Rewrite the given equation as y2=2xz3.

Substitute the value y2=2xz3 in the above distance equation.

Thus, d2=x2+2xz3+z2.

Let, f(x,y)=x2+2xz3+z2.

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

fx=x(x2+2xz3+z2)=x(x2)+x(2xz3)+x(z2)=2x+2z3(x2)+0=2x2x2z3

Hence, fx=2x2x2z3. (1)

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

fz=z(x2+2xz3+z2)=z(x2)+z(2xz3)+z(z2)=0+2x(3z4)+2z=6xz4+2z

Thus, fz=6xz4+2z

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