Question

Let $f$ and $g$ be two non-decreasing twice differentiable functions defined on an interval $(a,b)$ such that for each $x\in (a,b),f^n\left(x\right)=g\left(x\right)$ and $g^n\left(x\right)=f\left(x\right)$. Suppose also that f(x)g(x) is linear in $x$ on $(a,b)$. Show that we must have $f\left(x\right)=g\left(x\right)=0$ for all $x\in (a,b)$.

Answer 1

f and g \rightarrow double derivative exist (a,b)

$f^n\left(x\right)=g\left(x\right)$and

$g^n\left(x\right)=f\left(x\right)f^{\prime \prime }\left(x\right)=g\left(x\right)$

also

$f^{{}^{\prime }}\left(x\right)=g\left(x\right)\to f^{{}^{\prime }}\left(x\right)=f^{\prime \prime }\left(x\right)\Rightarrow f\left(x\right)=f^{{}^{\prime }}\left(x\right)+C\to g"\left(x\right)=f\left(x\right)$

also

$g^{\prime }\left(x\right)=f\left(x\right)g\left(x\right)=g^{\prime }\left(x\right)+C{f}^{\prime }\left(x\right)=g"\left(x\right)\Rightarrow f\left(x\right)=f^{\prime }\left(x\right)+Cf\left(x\right)=g"\left(x\right)+Cg"\left(x\right)=f\left(x\right)\Rightarrow C=0$

All these conditions only satisfy if $f\left(x\right)=g\left(x\right)=0$

Hence proved.