Question

A function f is thrice differentiable on [a,b] and f(a)=f(b)=0 and f'(a)=f'(b)=0. prove that f'''(c)=0 for some c belongs to (a,b).
use rolle's theorem.

Answer 1

Rolle's Thm: Suppose that y = f(x) is continuous at every point of the closed interval[a,b] and differentiable at every point of its interior (a,b).If
f(a) = f(b).
then there is at least one number c between a and b at which
f'(c) = 0.

Now for our case there must exsists a c different from a and b such that f'(c)=0
But that means there are three points such that f'(a)=f'(c)=f'(b)=0
Now we use Rolle's theorem again to find two points d and e that are between a and c, c and b respecfully, such that f''(d)=f''(e)=0

Now Rolle's theorem again to show there exists k such that f'''(k)=0 (proof done)

Answer 2

thanx