Question

The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.

विकल्प

• R

• `"R" - {1/2}`

• `(0, oo)`

• None of these

Answer 1

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}`