First, let's establish some definitions: f is said to be increasing on an interval I if for all x in I , f (x _{1}) < f (x _{2}) whenever x _{1} < x _{2} . f is said to be decreasing on an interval I if for all x in I , f (x _{1}) > f (x _{2}) whenever x _{1} < x _{2} . A function is monotonic on an interval I if it is only increasing or only decreasing on I .
The derivative can help us determine whether a function is increasing or decreasing on an interval. This knowledge will later allow us to sketch rough graphs of functions.
Let f be continuous on [a, b] and differentiable on (a, b) . If f'(x) > 0 for all x on (a, b) , then f is increasing on [a, b] . If f'(x) < 0 for all x on (a, b) , then f is decreasing on [a, b] .
This should make intuitive sense. In the graph below, wherever the slope of the tangent is positive, the function seems to be increasing. Likewise, wherever the slope of the tangent is negative, the function seems to be decreasing:
Example: Find regions where
f (x) =
x
^{3} -
x
^{2} - 6x
is increasing and
decreasing.
Solution:
f'(x) | = | x ^{2} - x - 6 | |
f'(x) | = | (x - 3)(x + 2) |
Now we find regions where f'(x) is positive, negative, or zero. f'(x) = 0 at x = 3 and x = - 2 . This can be marked on the number line: