Question

If f(x) = `{{:("m"x + 1",",  "if"  x ≤ pi/2),(sin x + "n"",",  "If"  x > pi/2):}`, is continuous at x = `pi/2`, then ______.

विकल्प

• m = 1, n = 0

• m = `("n"pi)/2 + 1`

• n = `("m"pi)/2`

• m = n = `pi/2`

Answer 1

If f(x) = `{{:("m"x + 1",",  "if"  x ≤ pi/2),(sin x + "n"",",  "If"  x > pi/2):}`, is continuous at x = `pi/2`, then n = `("m"pi)/2`.

Explanation:

Given that: f(x) = `{{:("m"x + 1",",  "if"  x ≤ pi/2),(sin x + "n"",",  "If"  x > pi/2):}`, is continuous at x = `pi/2`

L.H.L. = `lim_(x -> pi^-/2) ("m"x + 1)`

= `lim_("h" -> 0) ["m"(pi/2 - "h") + 1]`

= `("m"pi)/2 + 1`

R.H.L. = `lim_(x -> pi^+/2) (sinx + "n")`

= `lim_("h" -> 0) [sin(pi/2 + "h") + pi]`

= `lim_("h" -> 0) cos "h" + "n"`

= 1 + n

When f(x) is continuous  at x = `pi/2`

∴ L.H.L. = R.H.L.

`("m"pi)/2 + 1` = 1 + n

⇒ n = `("m"pi)/2`.