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Question

Find the values of a and b such that the function defined by f(x)=5, if x2ax+b, if 2<x<1021, if x10 is a continuous function.

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Solution

Here, f(x)= 5, if x2ax+b, if 2<x<1021, if x10

At x=2, LHL=limx2f(x)=limx2(5)=5

RHL = limx2+f(x)=limx2+(ax+b)

Putting x=2+h as x2+ when h0

limh0[a(2+h)+b]=limh0(2a+ah+b)=2a+b

Also, f(2)=5

Since, f(x) is continuous at x=2.

LHL = RHL = f(2) 2a+b=5 ....(i)

At x = 10, LHL=limx10f(x)=limx10(ax+b)

Putting x=10-h as x10 when h0

limh0[a(10h)+b]=limh0(10a+ah+b)=10a+b

RHL = limx10+f(x)=limx10+(21)=21

Als, f(10)=21

Since, f(x) is continuous at x=10.

LHL=RHL =f(10) 10a+b=21 ......(ii)

Subtracting Eq. (i) from Eq (ii), we get 8a = 16 a=2

Put a=2 in Eq. (i), we get 2×2+b=5b=1


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