Question

help: please correct my proof

Answer 1

I don't know but I think you shouldn't assume that f is differentiable. I suspect this can be solved using intermediate value theorem only.

Answer 2

Let \(g(z)=cf(x)+kf(y)-(c+k)f(z)\). \(g(z)\) is continuous because it is a combination of the continuous function \(f(x)\).

Suppose \(f(x)<f(y)\). Then \(g(y)<0\) and \(g(x)>0\). By Intermediate Value Theorem, there exist a \(\zeta\) such that \(x<\zeta<y\) and \(g(\zeta)=0\). \(g(\zeta)=0\implies f(\zeta)=\dfrac{cf(x)+kf(y)}{c+k}\).

The case of \(f(y)<f(x)\) and \(f(x)=f(y)\) is left as an exercise for the reader. The case of \(f(x)=f(y)\) is slightly harder, and that will be my belated gift for you on April Fool's day.