Published by:
CGP EDU Academic Team
Published on: August 12, 2026
If the function ƒ(x) and g(x) are continuous in [a, b] and differentiable in (a, b), then the equation 
=
(b –
A
has in the interval [a, b] - At least one root
B
Exactly one root
C
At most one root
D
No root
Text Solution
Verified by ExpertsThe correct answer is:
A
Let h(x) = 
= ƒ g(x) – g(a) ƒ(x).
Then, h ′ (x) = ƒ g ′ (x) – g(a) ƒ ′ (x) = 
Since ƒ (x) and g (x) are continuous in [a, b] and differentiable in (a, b), therefore, h (x) is also continuous in [a, b] and differentiable in (a, b). So, by Mean Value theorem, there exists at least one real number c, a < c < b for which
h ′ = 
,
∴ h(b) – h(a) = (b – a) h ′ … (1)
Here h(a) = 
= 0, h(b) = 
∴ from (1), 
= (b – a) h ′
= (b – a) 
.
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