I need help with the following question: $f,g: U \rightarrow R$ twice differentiable in $U \subset R^{n}$ conex open. If $f(a)=g(a)$, $df(a)=dg(a)$ and $d^{2}f(x)=d^{2}g(x), \forall x \in U$ so $f=g$.
I tried to use taylor's formula as follows: $f(a+v) = f(a) + df(a)\cdot v+ \frac{1}{2}d^{2}f(a)\cdot v^{2} + r_{f}(v)$ and the same for $g$, but can't get a relationship between $r_{f}$ and $r_{g}$, please... any tips are welcome
A hint:
Assume $a=0$, and put $f-g=: h$. Consider an arbitrary point $p\in U$. In order to prove $h(p)=0$ consider the auxiliary function $$\phi(t):=h(t\,p)\qquad(0\leq t\leq p)$$ and look at its Taylor expansion at $0$ (with remainder after the linear term). Use the chain rule to compute $\phi'(t)$, $\phi''(t)$.