Differentiable function/Equivalence relation by derivative condition/Exercise

Let ${}M$ be the set of all twice continuously differentiable functions from ${}\mathbb {R}$ to ${}\mathbb {R}$. Define on ${}M$ a relation by

$f\sim g{\text{ if }}f(0)=g(0),\,f'(0)=g'(0){\text{ and }}f^{\prime \prime }(1)=g^{\prime \prime }(1).$

a) Show that this is an equivalence relation.

b) Find for every equivalence class of this equivalence relation a polynomial representative.

c) Show that this equivalence relation is compatible with the addition of functions.

d) Show that this equivalence relation is not compatible with the multiplication of functions.

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