Rolle's theorem with a supplementary condition I have to find the types of roots (i.e real or complex) of the equation $$ 11^x + 13^x+ 17^x -19^x = 0 \dots (1) $$
If $$ f(x) = 11^x + 13^x+ 17^x -19^x = 0 $$ , then obviously $ f'(x)= 0 $ has a 0 solution, and indeed every derivative of $f(x)$ has a  0 solution. 
In this context a question arises in my mind : if all the conditions of Rolle's Theorem are satisfied for a function $g(x) $ in $[a,b]$, and in addition if $g'(c)=0$ ,then is it necessary that $c$ lies between $a$ and $b$ ? 
If it's true, then we can conclude that $f(x)=0 $
has more than 2 real roots right?
Any insight ? Thank you.


if all the conditions of Rolle's Theorem are satisfied for a function $g(x) $ in $[a,b]$, and in addition if $g'(c)=0$ ,then is it necessary that $c$ lies between $a$ and $b$ ? 

No, not at all. Consider for example, $g(x)=\sin(x)$ with $a=0$, $b=\pi$, and $c=\frac32\pi$.
Or (if you mean "strictly between $a$ and $b$"), consider $g(x)=\sin^2(x)$ with $a=0$ and $b=c=\pi$.
