If f(x) is continuous in [a, b] such that f(a) = b and f(b)…

Maths Functions, Limits, Continuity and Differentiability General Subjective Type

Published on: August 14, 2026

If f(x) is continuous in [a, b] such that f(a) = b and f(b) = a, then prove that there exists at least one c ∈ (a, b) such that f(c) = c.

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The correct answer is:

CHECK THE SOLUTION.

Let F(x) = f(x) – x

F(a) = f(a) – a = b – a

F(b) = f – b = a – b

F(x) is continous and F & F are of opposite signs.

∴ F(x) = 0 has atleast one root in (a, b)

f (x) – x = 0 ⇒ f (x) = x for atleast one c ∈ (a, b)

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