Read it again any time, including while you work through the exercises.
Look at the profile of a mountain stage: the road climbs, reaches a pass, then goes back down. A function curve is read exactly like that, always from left to right, in reading order. Describing these climbs and descents means studying the variations of the function.
The vocabulary is simple. On an interval, a function is increasing if, when x increases, f(x) increases too: the curve rises. It is decreasing if, when x increases, f(x) decreases: the curve falls. The highest point of a bump is called a maximum, the lowest point of a dip a minimum.
To summarise all this at a glance, we use a variation table: one row for x with the key values, one row for f with arrows. An upward arrow means increasing, a downward arrow means decreasing. At the tips of the arrows we write the values of f, which directly reveals the maxima and minima.
The sign of a function is a different question, not to be confused with variations. The function is positive when its curve is above the x-axis (f(x) > 0), negative when it is below (f(x) < 0), and it vanishes when the curve touches or crosses that axis.
A classic trap to defuse right away: a function can perfectly well go down while staying positive, like a plane losing altitude but still flying above the ground. Decreasing does not mean negative, and increasing does not mean positive. Variations describe the motion, the sign describes the position relative to the axis.
A function f is defined on [-3, 4]. Its curve goes down from the point (-3, 5) to the point (1, -2), then climbs back up to the point (4, 3). Describe the variations of f.
- 1
Lecture de gauche à droite
I travel along the curve from left to right, like a sentence, spotting where it rises and where it falls.
- 2
De x = -3 à x = 1 : f(x) passe de 5 à -2
On this part the curve falls: as x increases, f(x) decreases. So f is decreasing on [-3, 1].
- 3
De x = 1 à x = 4 : f(x) passe de -2 à 3
Then the curve climbs: f is increasing on [1, 4].
- 4
Minimum : f(1) = -2
The dip is reached where the direction changes. In the variation table: an arrow going down from 5 to -2, then an arrow going up from -2 to 3.
Result: f is decreasing on [-3, 1], increasing on [1, 4], and its minimum is -2, reached at x = 1.
On [-4, 5], the curve of f crosses the x-axis at x = -1 and x = 3. It is above the axis before -1 and after 3, below it in between. Build the sign table of f.
- 1
f(x) = 0 pour x = -1 et x = 3
The function vanishes where the curve touches the x-axis. These values split the interval into three zones.
- 2
Sur [-4 ; -1[ : courbe au-dessus, f(x) > 0
Above the axis, the values of f are positive.
- 3
Sur ]-1 ; 3[ : courbe en dessous, f(x) < 0
Between -1 and 3, the curve dips below the axis: f is negative.
- 4
Sur ]3 ; 5] : courbe au-dessus, f(x) > 0
After 3, the curve goes back above: f becomes positive again. The sign table therefore shows: +, 0, -, 0, +.
Result: f is positive on [-4, -1], negative on [-1, 3], positive on [3, 5], and vanishes at -1 and 3.