max f$( (a,b) )$= max $f( [a,b] )$ in continuous function

Question

Let $f:[a,b]→\mathbb{R}$ be a continuous function and gets its maximum in $(a,b)$. Prove that max $f( (a,b) )$= max $f( [a,b] )$

Answer 1

Let $c \in (a,b)$ be the point where $f$ attains its maximum in $(a,b)$. Assume $f(a)>\max(f((a,b)))$ (i.e. $f(a)=\max(f((a,b)))+d$ for some $d>0$), and then use the Intermediate Value Theorem on the interval $[a,c]$ to find a contradiction.

Answer 2

Assume hat $f(a)>\max_{x\in(a,b)}f(x)=f(x_0)$ for some $x_0\in(a,b)$.

Such $x_0$ exists since $f$ gets its maximum at $(a,b)$.

Then the continuity of $f$ at $a$ assures that we can find some $x_1\in(a,b)$ satisfying $f(x_1)>f(x_0)$.

So the assumption leads to a contradiction and we conclude that $f(a)\leq\max_{x\in(a,b)}f(x)$.

Same story for $b$.