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If f(a-x)=f(a+x) " and " f(b-x)=f(b+x) f...

If `f(a-x)=f(a+x) " and " f(b-x)=f(b+x)` for all real x, where `a, b (a gt b gt 0)` are constants, then prove that `f(x)` is a periodic function.

Text Solution

Verified by Experts

`f(x)=f(b+(x-b))`
` =f(b-(x-b))`
`=f(2b-x)`
`=f(a+(2b-x-a))`
`=f(a-(2b-x-a))`
`=f(2a-2b+x)`
Hence, `f(x)` is periodic with period `2a-2b.`
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