MHB Function Help - Rolle's Theorem or the Mean Value Theorem?

[vickon]

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?

 

[Opalg]

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?

Yes, you should use Rolle's theorem twice: first for the function $f$ and then for the function $f'$.

 

[SatyaDas]

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?

This problem just needs straight forward application of Rolle's Mean Value Theorem as already mentioned by the problem poster.