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The value of a and b such that the funct...

The value of a and b such that the function
`f(x)={(-2sinx"," , -pi le x le -(pi)/(2)),(a sinx+b",", -(pi)/(2) lt x lt (pi)/(2)),(cosx",", (pi)/(2) le x le pi ):}` is continuous in `[-pi,pi]` are

A

`-1, 0`

B

1, 0

C

1, 1

D

`-1, 1`

Text Solution

Verified by Experts

The correct Answer is:
D
For continuity in `[-pi, pi]`, we must have
At `x=-(pi)/(2),`
`f(-(pi)/(2))=lim_(x to (-(pi)/(2))^(-1))(-2 sinx)=lim_(x to (-(pi)/(2))^(+))(a sin x+b)`
`rArr 2= -a+b " " `…(i)
At `x=(pi)/(2),f((pi)/(2))=lim_(x to ((pi)/(2))^(-)) (a sinx+b) =lim_(x to ((pi)/(2))^(+))(cosx)`
`rArr 0=a+b " " ` ...(ii)
On solving Eqs. (i) and (ii), we get
a = -1 and b = 1
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  • The value of a and b such that the function f(x) {:{(-2 sin x", "-pile x le -pi/2 ),(a sin x+b " ," -pi/2 lt x lt pi/2 ),(cos x", " pi/2 le x le pi ):} is continuous in [ - pi , pi ] are

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