Question

let f be continuous on the interval [ 0,1] to R and such that f(0) = f(1). Prove that there exists a point c in [0,.5] such that f(c) = f(c+.5) let f be continuous on the interval [ 0,1] to R and such that f(0) = f(1). Prove that there exists a point c in [0,.5] such that f(c) = f(c+.5) @Mathematics

Answer 1

just use the IVT

Answer 2

Define g(x)=f(x)-f(x+.5). Then g(0)=f(0)-f(.5) and g(.5)=f(.5)-f(1). Let f(0)=f(1)=d and f(.5)=e. Then we have g(0)=d-e and g(.5)=e-d=-g(0). If g(0)=0 then the proof is complete, so suppose g(0) is not zero; say g(0)=u where u is nonzero. Then g(0) and g(.5) have opposite signs on [0, .5]. So, by the Intermediate Value Theorem, there is a c such that g(c)=0 on [0, .5]. But this implies that there is a c on [0, .5] such that f(c)=f(c+.5), and the proof is complete.

Answer 3

thanks but i get that g(c) = k. can i assume that k = 0 since g(0) = -g(.5)?