(cod: P-64-24-11) Consider the following statements:

(1) If \(f\) is a differentiable function on an open interval \(I\), with \(f'(x) > 0\) for all \(x \in I\), then \(f\) is increasing on \(I\).

(2) If \(x_0\) is a critical point of \(f\), then \(f\) has a local maximum or a local minimum at \(x_0\).

(3) Let \(f:I \subset \mathbb{R}\) be a continuous function on an open interval \(I\). Suppose there exists an interval \((a,b)\) such that \(x_0 \in (a,b) \subset I\) with \(f'(x)<0\) for all \( x \in (a,x_0)\) and \(f'(x)<0\) for all \(x \in (x_0,b)\). In this case, we can state that \(x_0\) is a local maximum point.

(4) If \(f\) is a twice-differentiable function such that \(f''(x) >0\) for all \(x\) in an open interval \(I\), then \(f'\) is increasing on \(I\). That is, \(f\) is concave up on \(I\), so its graph lies below all its tangents on \(I\).

(5) If \(f''(x_0) = 0\), then \((x_0,f(x_0))\) is an inflection point of the graph of \(f\).

(6) Suppose \(f'(x_0) = 0\) and that \(f\) is twice differentiable at \(x_0\). If \(f''(x_0) > 0\), then \(f\) has a local maximum at \(x_0\).

Regarding these statements, we have that: