Rolle's Theorem Proof

Proof: Assume:

1. Assume $f$ is continuous on $[a,b]$
2. Assume $f$ is differentiable on $(a,b)$
3. $f(a)=f(b)$
4. Suppose $f(a)=f(b)=k$ for some $k\in R$
5. Case (a):
   1. Suppose $f(x)=k$ for all $x\in (a,b)$
   2. Then $f^{\prime }(x)=0$ for all $x\in (a,b)$ derivative constant rule
   3. Then $\exists c\in (a,b)$ with $f^{\prime }(c)=0$ 2, ext gen
6. Case (b):
   1. Suppose $\exists x\in (a,b)$ s.t $f(x)>f(a)=f(b)$
   2. Then, by EVT, there exists a $c\in (a,b)$ s.t $f(c)$ is a local maxima/minima
      1. Suppose $f(c)$ is a local maxima
         1. Then, note that $f^{\prime }(c)$ exists as f is differentiable at $c\in (a,b)$
         2. By fermat’s theorem, $f^{\prime }(c)=0$
         3. Thus, $\exists c\in (a,b)$ with $f^{\prime }(c)=0$
      2. Suppose $f(c)$ is a local minima
         1. Then, note that $f^{\prime }(c)$ exists as f is differentiable at $c\in (a,b)$
         2. By fermat’s theorem, $f^{\prime }(c)=0$
         3. Thus, $\exists c\in (a,b)$ with $f^{\prime }(c)=0$
   3. Thus, $\exists c\in (a,b)$ with $f^{\prime }(c)=0$
7. Thus, $\exists c\in (a,b)$ with $f^{\prime }(c)=0$