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The function f defined by f(x) - x [1 + 1/3 sin (log x2)], a ≠ 0,/(0) = 0 ([] represents the greatest integer function) is
- a)continuous and differentiable at origin
- b)not continuous but differentiable
- c)continuous but not differentiable
- d)not continuous and not differentiable
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The function fdefined by f(x) - x[1 + 1/3 sin (log x2)], a ≠0,/(0) ...
and sin (-∞) can be any value between - 1 and 1
The value of integral part can also become ∞, but in all cases
due to factor x in it.f'(x) does not exist because
f' (x) for x is 1 but integral value which is its coefficient changes to give different values.
Most Upvoted Answer
The function fdefined by f(x) - x[1 + 1/3 sin (log x2)], a ≠0,/(0) ...
and sin (-∞) can be any value between - 1 and 1
The value of integral part can also become ∞, but in all cases
due to factor x in it.f'(x) does not exist because
f' (x) for x is 1 but integral value which is its coefficient changes to give different values.
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The function fdefined by f(x) - x[1 + 1/3 sin (log x2)], a ≠0,/(0) ...
Continuity and Differentiability of the Given Function
The function f(x) is defined as f(x) = x[1 + 1/3 sin(log(x^2))], where x ≠ 0 and f(0) = 0. We need to determine if the function is continuous and differentiable at the origin (x = 0).
Continuity
For a function to be continuous at a point, the limit of the function as x approaches that point must exist and be equal to the function's value at that point.
At x = 0, the function f(x) becomes f(0) = 0[1 + 1/3 sin(log(0^2))] = 0. Therefore, the function is continuous at x = 0.
Differentiability
For a function to be differentiable at a point, the derivative of the function at that point must exist.
To find the derivative of f(x), we need to consider the different cases of x separately due to the piecewise definition of the function. The derivative of f(x) will not exist at x = 0 because the term sin(log(x^2)) introduces a discontinuity at x = 0, making the function not differentiable at that point.
Therefore, the function f(x) is continuous at x = 0 but not differentiable at x = 0. The correct answer is option 'C' - continuous but not differentiable.
The function f(x) is defined as f(x) = x[1 + 1/3 sin(log(x^2))], where x ≠ 0 and f(0) = 0. We need to determine if the function is continuous and differentiable at the origin (x = 0).
Continuity
For a function to be continuous at a point, the limit of the function as x approaches that point must exist and be equal to the function's value at that point.
At x = 0, the function f(x) becomes f(0) = 0[1 + 1/3 sin(log(0^2))] = 0. Therefore, the function is continuous at x = 0.
Differentiability
For a function to be differentiable at a point, the derivative of the function at that point must exist.
To find the derivative of f(x), we need to consider the different cases of x separately due to the piecewise definition of the function. The derivative of f(x) will not exist at x = 0 because the term sin(log(x^2)) introduces a discontinuity at x = 0, making the function not differentiable at that point.
Therefore, the function f(x) is continuous at x = 0 but not differentiable at x = 0. The correct answer is option 'C' - continuous but not differentiable.
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The function fdefined by f(x) - x[1 + 1/3 sin (log x2)], a ≠0,/(0) = 0 ([] represents the greatest integer function) isa)continuous and differentiable at originb)not continuous but differentiablec)continuous but not differentiabled)not continuous and not differentiableCorrect answer is option 'C'. Can you explain this answer?
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