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Question

Let f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1sin3x3cos2x, if x<π2a, if x=π2b(1sinx)(π2x)2, if x>π2. If f(x) is continuous at x=π2, find a and b.

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Solution

Given,
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1sin3x3cos2x,if x<π2a,if x=π2b(1sinx)(π2x)2,if x>π2

f(π2)=a

LHL at x=π2, we have

limxπ2f(x)

=limh0f(π2h)

=limh01sin3(π2h)3cos2(π2h)

=13limh01cos3hsin2h

=13limh0(1cosh)(1+cos2h+cosh)(1cosh)(1+cosh)

=13×32=12

RHL at x=π2, we have

limxπ2+f(x)

=limh0f(π2+h)

=limh0b(1sin(π2+h))(π2(π2+h))2

=limh0b(1cosh)(2h)2

=limh02 bsin2(h2)16(h24)

=b8limh0⎜ ⎜ ⎜sinh2h2⎟ ⎟ ⎟2

=b8×1=b8

If f(x) is continuous at x=π2, then
limxπ2f(x)=limxπ2+f(x)=f(π2)

12=b8=a

b8=12b=4

a=12

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