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Question

Find the values of a and b so that the function f given by
fx=        1 ,if x3ax+b ,     if 3<x<5        7 ,if x5is continuous at x = 3 and x = 5.


Solution

Given: fx=1, if x3ax+b, if 3<x<57, if x5

We have
(LHL at x = 3) = limx3-fx=limh0f3-h=limh01=1                     

(RHL at x = 3) = limx3+fx=limh0f3+h=limh0a3+h+b=3a+b

(LHL at x = 5) = limx5-fx=limh0f5-h=limh0a5-h+b=5a+b                       

(RHL at x = 5) = limx5+fx=limh0f5+h=limh07=7

If f(x) is continuous at x = 3 and 5, then 

∴ ​limx3-fx =lim  x3+fx    and    limx5-fx =limx5+fx
1=3a+b    ...1  and  5a+b=7    ...2

On solving eqs. (1) and (2), we get
a=3 and b=-8

Mathematics
RD Sharma XII Vol 1 (2015)
Standard XII

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