Answer
See below.
Work Step by Step
1. A function $f(x)$ that is continuous on a closed interval $[a,b]$, has a local maximum at $x=x_0$. This is possible only when $f(x_0)$ is greater than all the enclosing values of the original function $f(x)$.
2. A function $f(x)$ that is continuous on a closed interval $[a,b]$, has a local minimum at $x=x_0$. This is possible only when $f(x_0)$ is less than all the enclosing values of the original function $f(x)$.
3. A function $f(x)$ that is continuous on a closed interval $[a,b]$, has a absolute maximum at $x=x_0$. This is possible only when $f(x_0)$ is greater or equal to all the enclosing values of the original function $f(x)$ and the value of $f(x_0)$ is an absolute maximum value of $f(x)$.
4. A function $f(x)$ that is continuous on a closed interval $[a,b]$, has a absolute minimum at $x=x_0$. This is possible only when $f(x_0)$ is less than or equal to all the enclosing values of the original function $f(x)$ and the value of $f(x_0)$ is an absolute minimum value of $f(x)$.
