Relation of functions satisfying rolle's theorem

Question

Let a > 1 and f,g: [-a,a] $\rightarrow$ R be twice differentiable functions such that for some c with 0 < c < 1 < a,

f(x) = 0 only for x=-a, 0, a

f$'$(x) = 0 = g(x) only x = -1, 0, 1

g$'$(x) = 0 only for x = -c, c

Then which relation between f and g is possible:$$ f = g' \ or \ f' = g \ or \ f = -g' $$

Answer 1

With Rolle we get

$f'(1)=f'(-1)=0$ and $g'(c)=g'(-c)=0.$

Hence $f = g'$ or $ f = -g'$ is not possible.