   Chapter 14.6, Problem 68E

Chapter
Section
Textbook Problem

(a) Show that the function f(x, y) = xy 3 is continuous and the partial derivatives fx and fy exist at the origin but the directional derivatives in all other directions do not exist.(b) Graph f near the origin and comment on how the graph confirms part (a).

(a)

To determine

To show: the function f(x,y)=xy3 is continuous and to prove that the partial derivatives fx and fx exist at origin and directional derivatives in other direction don not exit.

Explanation

Given:

The function f(x,y) is as below,

f(x,y)=xy3

Calculation:

The function f(x,y) is continuous at point (x0,y0) , when f(x0,y0) , lim(x,y)(x0,y0)f(x,y) and lim(x,y)(x0,y0)f(x,y)=f(x0,y0) .

For the condition f(0,0) at origin,

f(0,0)=03=0

The limit at the origin, substitute xy3 for f(x0,y0) in below equation,

lim(x,y)(0,0)f(x,y)=lim(x,y)(x0,y0)f(x0,y0)=lim(x,y)(0,0)xy3

For the function f(x,y)=xy1/3 is continuous on 2 it is a composition of a polynomial and the cube root function, both the functions are continuous

The partial derivative fx exists at (x0,y0) ,

f(x0,y0)=limh0f(x0+h,y0)f(x0,y0)h

For the limit at the origin

f(0,0)=limh0f(0+h,0)f(0,0)h=limh0h×030h=limh0(0)=0

The partial derivative fx is equal to zero and exists at origin

(b)

To determine

To graph: The function f near the origin.

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