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Question
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
Options
R
`"R" - {1/2}`
`(0, oo)`
None of these
Solution
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is `"R" - {1/2}`.
Explanation:
Given that: f(x) = |2x − 1| sinx
Clearly, f(x) is not differentiable at x = `1/2`
R.H.L. = `"f'"(1/2) = lim_("h" -> 0) ("f"(1/2 + "h") - "f"(1/2))/"h"`
= `lim_("h" -> 0) (|2(1/2 + "h") - 1|sin(1/2 + "h") - 0)/"h"`
= `lim_("h" -> 0) (|2"h"| sin((1 + 2"h")/2))/"h"`
= `2 sin (1/2)`
Also L.H.L. = `"f'"(1/2) = lim_("h" -> 0) ("f"(1/2 - "h") - "f"(1/2))/(-"h")`
= `lim_("h" -> 0) (|2(1/2 - "h") - 1|[- sin (1/2 - "h")] - 0)/(-"h")`
= `(|-2"h"|[-sin(1/2 - "h")])/(-"h")`
= `- 2 sin (1/2)`
∴ R.H.L. = `"f'"(1/2)` ≠ L.H.L. `"f'"(1/2)`
So, the given function f(x) is not differentiable at x = `1/2`.
∴ f(x) is differentiable in `"R" - {1/2}`
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