# Find a and b, so that the function f(x) defined by - Mathematics and Statistics

Sum

Find a and b, so that the function f(x) defined by

f(x)=-2sin x,       for -π≤ x ≤ -π/2

=a sin x+b,  for -π/2≤ x ≤ π/2

=cos x,        for π/2≤ x ≤ π

is continuous on [- π, π]

#### Solution

f(x)=-2sinx, " for " -pi<=x<=-pi/2

=asinx+b , " for " -pi/2<x<pi/2

=cosx , " for " pi/2<=x<pi

f(x) is continuous for x=-π/2

RHL,

=lim_(x->-pi/2)asinx+b

=asin(-pi/2)+b

=-a+b

f(-pi/2)=-2sin(-pi/2)

therefore-a+b=2........(i) [because f(x) " is continuous for "x=-x/2]

f(x) " is continuous for "x=pi/2

LHL,

=lim_(x->pi/2)asinx+b

=asin(pi/2)+b

=a+b

f(pi/2)=cos(pi/2)=0

a+b=0   .........(ii)

Solving (i) and (ii)

a= -1 and b=1

Concept: Definition of Continuity - Continuity of a Function at a Point
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