For all twice differentiable functions f:RR, with f(0)=f(1)=f'(0)=0:

Question

For all twice differentiable functions $f:\mathbb{R}	o\mathbb{R}$, with $f(0)=f(1)=f'(0)=0$:

Accepted Answer

f : R $$ \to $$ R, with f(0) = f(1) = 0

and f'(0) = 0

$$ \because $$ f(x) is differentiable and continuous

and f(0) = f(1) = 0

Applying Rolle's theorem in [0, 1] for function f(x)

f'(c) = 0, c $$ \in $$ (0, 1)

Now again

$$ \because $$ f'(c) = 0, f'(0) = 0

again applying Rolles theorem in [0, c] for function f'(x)

f''(c1) = 0 for some c1 $$ \in $$ (0, c) $$ \in $$ (0, 1)

Solutions · 1