(cod: P-61-26-6) Consider the following statements:

(1) A continuous function \(f:[a,b] \to \mathbb{R}\) on a closed and bounded interval \([a,b]\) must attain an absolute minimum value.

(2) If a continuous function has a local extremum at a number \(x_0\) in its domain, then \(f\) must be differentiable at \(x_0\) and \(f'(x_0)=0\).

(3) If a continuous function has a critical point at a point \(x_0\) in the interior of its domain, then \(f\) has a local extremum at \(x_0\).

(4) If \(f:[a,b] \to \mathbb{R}\) is a continuous function on the compact interval (closed and bounded) \([a,b]\), then \(f\) attains an absolute maximum value. Furthermore, this value is either attained at an endpoint of the interval or at a critical point of \(f\) in \((a,b)\).

(5) Let \(f:I \to \mathbb{R}\) be a continuous function on an open interval \(I\) that has a critical point at \(x_0 \in I\). If \(f'(x)>0\) for all \(x < x_0\) in \(I\) and \(f'(x)<0\) for all \(x > x_0\) in \(I\), then \(f\) attains an absolute maximum at \(x_0\).

Among the statements above, we can assert that: