Suppose $f$ is continuous for $a \leq x \leq b$. Show that the function attains both a maximum and minimum value. That is, there are at least two numbers, $c_{1}$ and $c_{2},$ between $a$ and $b,$ such that, $f\left(c_{1}\right)=M$ and $f\left(c_{2}\right)=N,$ where $f(x) \leq M$ and $f(x) \geq N,$ for all $x$ such that $a \leq x \leq b .$ This is called the Extreme Value Theorem