The function Y=| 2-3x | a)is continuous x R and differentiable x Rb...
Continuity of the function:
To determine the continuity of the function Y = |2-3x|, we need to check if the function is continuous at every point in its domain, which is all real numbers.
The function Y = |2-3x| can be written as two separate functions depending on the value of (2-3x).
When (2-3x) ≥ 0, the function becomes Y = 2-3x.
When (2-3x) < 0,="" the="" function="" becomes="" y="-(2-3x)" =="" />
Case 1: (2-3x) ≥ 0:
In this case, Y = 2-3x.
The function Y = 2-3x is a linear function and is continuous for all real numbers.
Case 2: (2-3x) < />
In this case, Y = -(2-3x) = 3x-2.
The function Y = 3x-2 is also a linear function and is continuous for all real numbers.
Combining the two cases:
Since both cases result in linear functions that are continuous for all real numbers, we can conclude that the function Y = |2-3x| is continuous for all real numbers.
Differentiability of the function:
To determine the differentiability of the function Y = |2-3x|, we need to check if the derivative exists at every point in its domain.
The derivative of the function Y = |2-3x| can be calculated by applying the chain rule.
When (2-3x) ≥ 0, the derivative is -3.
When (2-3x) < 0,="" the="" derivative="" is="" />
Case 1: (2-3x) ≥ 0:
In this case, the derivative is -3. The function Y = |2-3x| is differentiable for all x where (2-3x) ≥ 0.
Case 2: (2-3x) < />
In this case, the derivative is 3. The function Y = |2-3x| is differentiable for all x where (2-3x) < />
Combining the two cases:
The function Y = |2-3x| is differentiable for all real numbers except for the points where (2-3x) = 0, which is x = 2/3.
Conclusion:
Based on the analysis above, we can conclude that the function Y = |2-3x| is continuous for all real numbers and differentiable for all real numbers except at x = 2/3. Therefore, the correct answer is option 'C'.
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