Definition

1.   Given any x₁  & x₂  from an interval  I with x₁ < x₂  if f ( x₁ ) < f ( x₂ ) then f ( x ) is increasing on I.

2.   Given any x₁  & x₂  from an interval  I with x₁ < x₂  if f ( x₁ ) > f ( x₂ ) then f ( x ) is decreasing on I.

This definition will in fact be utilized in the proof of the next fact in this section.

The Given fact summarizes up what we were doing in the previously

Fact

1.   If f ′ ( x ) = 0 for each x on some interval I, then f ( x) is increasing on the interval.

2.   If f ′ ( x ) = 0 for each x on some interval I, then f (x ) is decreasing on the interval.

3.   If f ′ ( x ) = 0 for each x on some interval I, then f ( x ) is constant on the interval.