I cannot find the correct answer

Find an inequality that relates the sength of the altitudes in a triangle (you may have to consider area and side lengths.

Let h be the third altitude.  Then by the triangle inequality, h + 12 > 14, h + 14 > 12, and 12 + 14 > h.  The largest integer h that satisfies these inequalities is h = 25.

Since the altitudes are inside the triangle, the triangle is acute.  This gives us the inequalities 12^2 + 14^2 > h^2, h^2 + 12^2 > 14^2, and h^2 + 14^2 > 12^2.  Then the largest h can be is 18.