Given-f-0-2-R-f-x-is-twice-derivable-and-f-0-f-1-f-2-0-i-Show-that-there-exist-c-1-c-2-such-that-f-c-1-0-and-f-c-2-0-ii-Show-that-there-exist-c-3-such-that-f-c-3-0-

Question Number 100388 by Ar Brandon last updated on 26/Jun/20

G iven f : [0, 2] \to R, f (x) is twice derivable and

f (0) = f (1) = f (2) = 0

i - S how that there exist c_{1}, c_{2}, such that f^{'} (c_{1}) = 0

and f^{'} (c_{2}) = 0

i i - S how that there exist c_{3} such that f^{″} (c_{3}) = 0

Answered by maths mind last updated on 26/Jun/20

f (0) = f (1) \Rightarrow \exists c \in] 0, 1 [, f^{'} (c_{1}) = 0

f (1) = f (2) \Rightarrow \exists c_{2} \in] 1, 2 [f^{'} (c_{2}) = 0

l e t f^{'} (x) o v e r [c_{1}, c_{2}]

f^{'} (c_{1}) = f^{'} (c_{2}) \Rightarrow \exists c_{3} \in [c_{1}, c_{2}] s u c h f^{″} (c_{3}) = 0

i u s e d i f f c o n t i n u s d i f f e r e n t i a b l o v e r [a, b] s u c h

f (a) = f (b) \Rightarrow \exists c \in [a, b] s u c h f^{'} (c) = 0 . R o l l t h e o r e m

Commented by DGmichael last updated on 26/Jun/20

��very good.

Commented by Ar Brandon last updated on 26/Jun/20

Thanks ��

Commented by Ar Brandon last updated on 26/Jun/20

Ouaye DG, dès que ce monsieur se connecte oulala !�� On dirait une machine qui venait d'être activée.��