Increasing and Decreasing functions
A function, f(x), is increasing on an interval [ a, b ] if
f ( x1 )  f ( x 2 )
x 1  x 2 in [a , b].
whenever
If f(x) is differentiable and increasing then it's derivative must be nonnegative; and if the
derivative is postive then f(x) is increasing.
Notice that the function
f ( x )  x is increasing on the whole real line.
3
A function, f(x), is decreasing on an interval [ a, b ] if
f ( x1 )  f ( x 2 )
x 1  x 2 in [a , b].
whenever
If f(x) is differentiable and decreasing then it's derivative must be nonpositive; and if the
derivative is negative then f(x) is decreasing.
To determine the intervals where a function is increasing or decreasing you can look at a
graph. Sometimes we only have the derivative of our function and then we look at it's
graph or make a sign chart for the derivative. See the notes on making sign chart
Local Maxima and Minima
Graph the function f ( x )  30 ( x  2 x  24 x ) . the top of the hill is called a
local max and the bottom of the valley is called a local min.
3
Now graph the derivative of this function.
2
The local max and min of f occur where the derivative of f changes sign.
To find these local max and min points we look for sign changes in the derivative. Where can these occur?
A function can only change sign if it crosses the x axis or a discontinuity. So f ' can change sign at its zeros
or at its discontinuities.
Where f ' changes from positive to negative, f changes from increasing to decreasing. If f is continuous at
such a point it has a local max.
Where f ' changes from negative to positive, f changes from decreasing to increasing. If f is continuous at
such a point it has a local min.
The word local indicates that this need not be the overall extreme value ( max or min) of the function. In
the above example the function becomes arbitrarily large negative on the left and arbitrarily large positive
on the right. That function has no absolute max or min on the whole real line.
Absolute Max and Min
A point (c, f(c)) is an absolute maximum on [ a, b ] if
A point (c, f(c)) is an absolute minimum on [ a, b ] if
f ( x )  f ( c ) for all x in [ a, b].
f ( x )  f ( c ) for all x in [ a, b].
Fermat's theorem If a function is continuous on a closed bounded interval [ a, b] then it attains an
absolute maximum and an absolute minimum on [ a, b ]. These must occur at the endpoints a or b or at
points where the derivative is 0 or where the derivative does not exist.
Examples: Find all local extrema. Find the absolute extrema on the given interval.
x
2
1.
f ( x )  xe
[  1, 2 ]
2.
f ( x)  x ( x  7)
3.
f ( x )  x ( x  1)
3
2
[  1,8 ]
4
3
Find the x-values of any local extrema for the function with the given derivative.
4.
f '( x)  x ( x  2)
5.
f ' ( x )  ( x  1) ( x  2 ) e
2
3
3
4
x