Mathematics Exam > Mathematics Questions > Letthena)f(x, y) is continuous at originb)f(x...
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Let
then
then
- a)f(x, y) is continuous at origin
- b)f(x, y) has removable discontinuity of (0,0)
- c)f(x, y) is not differentiable at (0, 0)
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Letthena)f(x, y) is continuous at originb)f(x, y) has removable discon...
We are given that
Let us approaches (0, 0) along the liney = mx, we get
which depends on m. Thus
does not exists and hence, f(x, y) is discontinuous at origin. Thus, option (a), (b) are incorrect.
Now,
So that, A = 0, B = 0 which does not depends on h and k and
Now, approaching (0, 0) along the line h = mk, we get
which depends on m and hence does not exists. Therefore , f(x, y) is not differentiable at origin. Hence, option (c) is correct.
Let us approaches (0, 0) along the liney = mx, we get
which depends on m. Thus
does not exists and hence, f(x, y) is discontinuous at origin. Thus, option (a), (b) are incorrect.Now,
So that, A = 0, B = 0 which does not depends on h and k andNow, approaching (0, 0) along the line h = mk, we get
which depends on m and hence does not exists. Therefore , f(x, y) is not differentiable at origin. Hence, option (c) is correct.
Most Upvoted Answer
Letthena)f(x, y) is continuous at originb)f(x, y) has removable discon...
ANSWER :- c
Solution :- Consider an approach along the line y=x; then f(x,y)=x^2/(x^2+x^2) = 1/2
for all x≠0. On the other hand, if we approach (0,0) along the line y=2x
f(x,y)=(2x)^2/[x^2+4x^2]
=2/5
So there are two different paths toward the origin, each giving a different limit. Hence, the limit does not exist.
Or more simply, approach along y=0
Therefore , f(x, y) is not differentiable at origin
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Letthena)f(x, y) is continuous at originb)f(x, y) has removable discontinuity of (0,0)c)f(x, y) is not differentiable at (0, 0)d)None of theseCorrect answer is option 'D'. Can you explain this answer?
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