Mathematics Exam > Mathematics Questions > Leta)f(x, y) is continuous at originb)f(x, y)...
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Let
- a)f(x, y) is continuous at origin
- b)f(x, y) is differentiable at origin
- c)fx(0,0) ≠ 0
- d)fy(0 ,0) ≠ 0
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at ori...
We are given that 
Let us take ε > 0 and x2 + y2 ≠ 0.
Now consider,
We know that,
or equivalently
Thus,
| f(x ,y )-0 | ≤ | y | = 0- |x| + 1 • |y|
take
Therefore, for given ε > 0, there exists δ > 0 such that
|f(x, y) - 0 | < ε. whenever 0<|x|<δ,
0< |y |< δ
Hence,
Thus, f(x,y) is continuous at (0, 0).
Hence, option (a) is correct.
Now,
f(0 + h ,0 + k) - f(0,0) = f(h , k) - f(*0,0)
= 0 • h + 0 • k +
• 
So that A = 0, B = 0 which does not depends on h and k and
Now approaching along h = mk, we get
which depends on m. Therefore
does not exists. Hence, f(x, y) is not differentiable at origin. Hence, option (b) is incorrect.
Next,
Hence, option (c) is incorrect.
And,
Hence, option (d) is incorrect.
Let us take ε > 0 and x2 + y2 ≠ 0.
Now consider,
We know that,
or equivalently
Thus,
| f(x ,y )-0 | ≤ | y | = 0- |x| + 1 • |y|
take
Therefore, for given ε > 0, there exists δ > 0 such that
|f(x, y) - 0 | < ε. whenever 0<|x|<δ,
0< |y |< δ
Hence,
Thus, f(x,y) is continuous at (0, 0).
Hence, option (a) is correct.
Now,
f(0 + h ,0 + k) - f(0,0) = f(h , k) - f(*0,0)
= 0 • h + 0 • k +
• So that A = 0, B = 0 which does not depends on h and k and
Now approaching along h = mk, we get
which depends on m. Therefore
does not exists. Hence, f(x, y) is not differentiable at origin. Hence, option (b) is incorrect.Next,
Hence, option (c) is incorrect.And,
Hence, option (d) is incorrect.
Most Upvoted Answer
Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at ori...
We are given that 
Let us take ε > 0 and x2 + y2 ≠ 0.
Now consider,
We know that,
or equivalently
Thus,
| f(x ,y )-0 | ≤ | y | = 0- |x| + 1 • |y|
take
Therefore, for given ε > 0, there exists δ > 0 such that
|f(x, y) - 0 | < ε. whenever 0<|x|<δ,
0< |y |< δ
Hence,
Thus, f(x,y) is continuous at (0, 0).
Hence, option (a) is correct.
Now,
f(0 + h ,0 + k) - f(0,0) = f(h , k) - f(*0,0)
= 0 • h + 0 • k +
• 
So that A = 0, B = 0 which does not depends on h and k and
Now approaching along h = mk, we get
which depends on m. Therefore
does not exists. Hence, f(x, y) is not differentiable at origin. Hence, option (b) is incorrect.
Next,
Hence, option (c) is incorrect.
And,
Hence, option (d) is incorrect.
Let us take ε > 0 and x2 + y2 ≠ 0.
Now consider,
We know that,
or equivalently
Thus,
| f(x ,y )-0 | ≤ | y | = 0- |x| + 1 • |y|
take
Therefore, for given ε > 0, there exists δ > 0 such that
|f(x, y) - 0 | < ε. whenever 0<|x|<δ,
0< |y |< δ
Hence,
Thus, f(x,y) is continuous at (0, 0).
Hence, option (a) is correct.
Now,
f(0 + h ,0 + k) - f(0,0) = f(h , k) - f(*0,0)
= 0 • h + 0 • k +
• So that A = 0, B = 0 which does not depends on h and k and
Now approaching along h = mk, we get
which depends on m. Therefore
does not exists. Hence, f(x, y) is not differentiable at origin. Hence, option (b) is incorrect.Next,
Hence, option (c) is incorrect.And,
Hence, option (d) is incorrect.
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Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer?
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Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer?.
Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer?.
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