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Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous?
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Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is ...
Function Definition
The function f(x) = |sin(πx)| is defined for all real numbers x. Now, let's analyze its properties regarding continuity and differentiability.
Continuity of f(x)
- Continuous Everywhere: The sine function is continuous everywhere, and since the absolute value function is also continuous, the composition of these functions, |sin(πx)|, is continuous for all x in R.
- Key Points: At points where sin(πx) = 0 (i.e., x = n, where n is an integer), the function transitions smoothly from positive to zero or zero to positive, maintaining continuity.
Differentiability of f(x)
- Differentiable Everywhere Except at Integer Values: The derivative of |sin(πx)| exists everywhere except at the points where sin(πx) = 0 (i.e., at integer x values).
- Key Points: At these points, the function has a sharp corner, leading to a non-differentiable condition. The left-hand and right-hand derivatives do not match.
Conclusion
Considering the above points, the correct characterization of the function f(x) = |sin(πx)| is:
- Continuous Everywhere: f(x) is continuous for all x in R.
- Differentiable at Non-Integer Values: It is differentiable everywhere except at integer values of x.
Thus, the correct option is (c): f is continuous everywhere and differentiable everywhere except at integral values of x.
The function f(x) = |sin(πx)| is defined for all real numbers x. Now, let's analyze its properties regarding continuity and differentiability.
Continuity of f(x)
- Continuous Everywhere: The sine function is continuous everywhere, and since the absolute value function is also continuous, the composition of these functions, |sin(πx)|, is continuous for all x in R.
- Key Points: At points where sin(πx) = 0 (i.e., x = n, where n is an integer), the function transitions smoothly from positive to zero or zero to positive, maintaining continuity.
Differentiability of f(x)
- Differentiable Everywhere Except at Integer Values: The derivative of |sin(πx)| exists everywhere except at the points where sin(πx) = 0 (i.e., at integer x values).
- Key Points: At these points, the function has a sharp corner, leading to a non-differentiable condition. The left-hand and right-hand derivatives do not match.
Conclusion
Considering the above points, the correct characterization of the function f(x) = |sin(πx)| is:
- Continuous Everywhere: f(x) is continuous for all x in R.
- Differentiable at Non-Integer Values: It is differentiable everywhere except at integer values of x.
Thus, the correct option is (c): f is continuous everywhere and differentiable everywhere except at integral values of x.
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Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous?
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Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous?.
Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = |sin pi*x| ,x \in R , (a) f is continuous nowhere (b) f is continuous everywhere and differentiable nowhere (c) f is continuous everywhere and differentiable everywhere except at integral values of x (d) f is differentiable everywhere but not continuous?.
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