   Chapter 14, Problem 5P

Chapter
Section
Textbook Problem

Suppose f is a differentiable function of one variable. Show that all tangent planes to the surface z = xf(y/x) intersect in a common point.

To determine

To show: All the tangent planes to the surface z=xf(yx) intersects in a common point.

Explanation

Given:

The equation of the surface is z=xf(yx) .

Calculation:

Let, g(x,y)=xf(yx) (1)

Differentiate equation (1) with respect to x using the product rule ddx(u.v)=dudxv+dvdxu .

gx(x,y)=f(yx)+xf'(yx)(yx2)=f(yx)x(yx2)f'(yx)gx(x,y)=f(yx)(yx)f'(yx)

Differentiate the equation (1) with respect to y .

gy(x,y)=(0)f(yx)+xf'(yx)(1x)=xf'(yx)(1x)=x(1x)f'(yx)gy(x,y)=f'(yx)

The general equation of the tangent plane at the point (x0,y0,z0) is, zz0=fx(x0,y0)(xx0)+fy(x0,y0)(yy0) .

Here, fx(x0,y0)=gx(x,y) and fy(x0,y0)=gy(x,y) .

Rearrange the general equation of tangent plane

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