Question

if f'(a)= g'(a), f'(b)= g'(b) Does c€(a,b) exist for f''(c)=g''(c) ? PS:f', g' €[a,b] f'', g'' €(a,b)

Answer 1

we are assuming everything is differentiable right? so forget about the primes, replace \(f'\) by say \(f\) and \(g'\) by \(g\) then use mean value theorem
\(f(a)=g(a)\) and \(f(b)=g(b)\)
we know there is a \(c\in (a,b)\)with \(f'(c)=\frac{f(b)-f(a)}{b-a}\) and
\(c\in (a,b)\)with \(g'(c)=\frac{g(b)-g(a)}{b-a}\)

since \(\frac{f(b)-f(a)}{b-a}=\frac{g(b)-g(a)}{b-a}\) the result follows

Answer 2

all of this presupposed that \(f\) and \(g\), (or in your case \(f'\) and \(g'\) ) are differentiable, otherwise all bets are off

Answer 3

vi know that f' g' is continious on [a,b] and f'' g'' exists on (a,b)