Question

If f(x) is continuous over [- π, π], where f(x) is defined as

f(x) = `{(- 2 sin x ","  - pi ≤ x ≤ (-pi)/2),(alpha sin x + beta","  - pi/2 < x < pi/2),(cos x ","   pi/2 ≤ x ≤ pi):}` then α and β equals 

विकल्प

• α = - 1, β = 1

• α = 1, β = - 1

• α = 1, β = 1

• α = β = 0

Answer 1

α = - 1, β = 1

Explanation:

Since, f(x) is continuous over [- π, π].

∴ It ts continuous at x = `- pi/2 and x = pi/2`.

∴ `lim_(x -> (-pi^-)/2) "f"(x) = lim_(x -> (-pi^+)/2) "f"(x)`

⇒ `lim_(x -> (-pi)/2) (- 2 sin x) = lim_(x -> (-pi)/2) (alpha sin x + beta)`

⇒ - 2(- 1) = α(- 1) + β

⇒ - α + β = 2       ...(i)

Also, `lim_(x -> (-pi^-)/2) "f"(x) = lim_(x -> (-pi^+)/2) "f"(x)`

⇒ `lim_(x -> (pi)/2) (alpha sin x + beta) = lim_(x -> (pi)/2) (cos x)`

⇒ α(1) + β = 0 

⇒ α + β = 0      ....(ii)

From (i) and (ii), we get

α = - 1, β = 1