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

Toolbox:
• Let $f:[a,b]\rightarrow R$ be continuous on [a,b] and differentiable on (a,b).Such that $f(a)=f(b)$ where a and b are some real numbers.Then there exists some $c$ in $(a,b)$ such the $f'(c)=0$
Step 1:
For Rolle's theorem if
(i) $f$ is continiuous in $[a,b]$
(ii) $f$ is derivable in $[a,b]$
(iii) $f(a)=f(b)$
Step 2:
$f'(c)=0$$\quad c\in (a,b)$
We are given $f$ is continuous and derivable.
But $f'(c)\neq o\Rightarrow f(a)\neq f(b)$
$f'(-5)\neq f(5)$