(a) The function f ( x , y ) = x y x 2 + y 2 if ( x , y ) ≠ ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) was graphed in Figure 4. Show that f x ( 0 , 0 ) and f y ( 0 , 0 ) both exist but f is not differentiable at ( 0 , 0 ) . [Hint: Use the result of Exercise 53.] (b) Explain why f x and f y are not continuous at ( 0 , 0 ) .
(a) The function f ( x , y ) = x y x 2 + y 2 if ( x , y ) ≠ ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) was graphed in Figure 4. Show that f x ( 0 , 0 ) and f y ( 0 , 0 ) both exist but f is not differentiable at ( 0 , 0 ) . [Hint: Use the result of Exercise 53.] (b) Explain why f x and f y are not continuous at ( 0 , 0 ) .
Solution Summary: The author explains that the function f isn't differentiable at (0,0).
f
(
x
,
y
)
=
x
y
x
2
+
y
2
if
(
x
,
y
)
≠
(
0
,
0
)
0
if
(
x
,
y
)
=
(
0
,
0
)
was graphed in Figure 4. Show that
f
x
(
0
,
0
)
and
f
y
(
0
,
0
)
both exist but
f
is not differentiable at
(
0
,
0
)
. [Hint: Use the result of Exercise 53.]
(b) Explain why
f
x
and
f
y
are not continuous at
(
0
,
0
)
.
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