For any x we have -|x| <= x <= |x|. So, -f''(a+h) <= |-f''(a+h)| = |f''(a+h)|. Further, |f''(a+h)| <= max[abs(f''(x))] between a and (a+h) because the left-hand side is just one element of the set over which the maximum is taken (this set is {|f''(x)| : a <= x <= a + h}). Therefore, -f''(a+h) <= max[abs(f''(x))]. Similarly, -|f''(a+h)| = -|-f''(a+h)| <= -f''(a+h) and -max[abs(f''(x))] <= -|f''(a+h)|, so -max[abs(f''(x))] <= -f''(a+h).