Let f : R → R be continuous and periodic with period T > 0 . then

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Let $f : R \to R$ be continuous and periodic with period $T > 0$ . then

Answer:

Solution:

Let $g (x) = f (x + T /2) - f (x)$ .

Then $g (k) = f (k + T /2) - f (k)$ ........(i)

and $g (k + T /2) = f (k + T) - f (k + T /2) = f (k) - f (k + T /2) = - g (k)$ .

Hence, by the Intermediate Value Theorem, there exists an $x_{0} \in [k, k + T /2]$ for which $g (x_{0}) = 0$ .

Stream:JEESubject:MathematicsTopic:FunctionsSubtopic:Periodic Functions

⏱ 2mℹ️ Source: QB

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