Let f(x) = 
Find the values of the constants a, b, p and q so that
(i) f(x) is continuous for all x.
(ii) f(x) is not differentiable at x = 1
(iii) f ′ (x) is continuous at x = 3
Text Solution
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Sol. It is given that f(x) is everywhere continuous. So, it is continuous at x = 1 and x = 3.
∴ 
f(x) = 
f(x) = f(1) and 
f(x) = 
f(x)=(3)
⇒ 
ax (x –1) + b = 
x –1
and,
x –1 = 
px 2 + qx + 2
⇒ b = 0 and 2= 9p + 3q + 2
⇒ b = 0 and 9p + 3q = 0
⇒ b = 0 and 3p + q = 0
Now, f(x) is not differentiable at x = 1
⇒ (LHD at x = 1) ≠ (RHD at x = 1)
⇒ 
≠ 
⇒ [a (2x –1) x = 1 ≠ 1
⇒ a ≠ 1
It is given that f ′ (x) is continuous at x = 3.
∴ 
1 = 
(2px + q)
⇒ 1 = 6p + q
Solving (i) and (ii), we get
p = 
and q = –1
Hence, a ≠ 1, b = 0, p = 
and q = –1
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