Hi fmunshi!!
Let me call the open interval where f is twice differentiable (p,q),
where:
p < x1 < x2 < x3 < q (I choose x1 < x2 < x3 just for convenience).
We will use the Rolle´s Theorem:
If f is continuous on [a, b] ,differentiable on (a, b), and f(a) =
f(b) = 0, then there exists a number c in (a, b) such that f'(c) = 0.
For the proof of this theorem see:
http://pirate.shu.edu/projects/reals/cont/proofs/rollethm.html
Case A)
We have x1 < x2 and f(x1) = f(x2) = 0.
p < x1 < x2 < x3 < q ;
then x1 and x2 are in the interval (p,q) where f is twice
differentialble, then f is continuous in [x1,x2] , differentiable in
(x1,x2) and
f(x1) = f(x2) = 0 ,
then the Rolle´s theorem is applicable to this case and exists a
number c1 in (x1,x2) such that f´(c1) = 0.
Case B)
In the same way we can use the Rolle´s theorem to say that there is a
number c2 in (x2,x3) such that f´(c2) = 0.
Let be g the derivative function of f: g(x) = f´(x), then g is
differentiable in (p,q) (because f is twice differentiable) and both
c1 and c2 are numbers in (p,q) such that g(c1) = g(c2) = 0.
Here the Rolle´s theorem is applicable to g in the interval [c1,c2].
Then exist a number z in (c1,c2) such that g´(z) = f´´(z) = 0. And the
problem is solved.
I hope this helps. Please if you need a clarification post a request
for it.
Best regards.
livioflores-ga |
