If $f:[ - 5,5] \to R$ is a differentiable function and if $f'(x)$does not vanish anywhere, then prove that $f( - 5) \ne f(5)$.

For Rolle's theorem, if

(i) f is continuous in [a, b]

(ii) f is derivable in (a, b)

(iii) $f(a) = f(b)$
then $f'(c) = 0,c \in (a,b)$

We are given/is continuous and derivable but
$f'(c) \ne 0 \Rightarrow f(a) \ne f(b)$ i.e. $f( - 5) \ne f(5)$
Hence proved.