CURVE SKETCHING
(Increasing and Decreasing Functions)
One of the most useful things to know about a function is where its graph rises and
where it falls (always reading the graph from LEFT to RIGHT).
Consider the given function:
A
B
C
D
1.




1
3
6
8
Using interval notation, state where the function is:
a)
increasing
____________________
b)
decreasing
____________________
2.
Draw tangent lines (on the graph) in each of the above intervals.
3.
What do you notice about the slope of the tangents (ie) f ’(x) in the:
a)
increasing intervals?
____________________
b)
decreasing intervals?
____________________
c)
max/min points?
____________________
For a function, f(x), that is continuous and differentiable on an interval:
 f(x) is increasing if f ( x 1 )  f ( x 2 ) for any x 1  x 2 in the interval
 f(x) is decreasing if f ( x 1 )  f ( x 2 ) for any x 1  x 2 in the interval
For functions that are both continuous and differentiable, the intervals of increase and
decrease can be found using the derivative (slopes of the tangents) of the function.
FIRST DERIVATIVE TEST:
For a function, f(x), that is continuous and differentiable on an interval:
 If f ' ( x )  0 , then f ( x ) is increasing on the interval.
 If f ' ( x )  0 , then f ( x ) is decreasing on the interval.
 If f ' ( x )  0 , then x is a possible local maximum or minimum value.
f '(x)0
f '(x)0
f '(x)0
Ex 
Determine the intervals of increase and decrease for each function:
a)
f( x ) = x3 + 3x2 – 2
f ‘(x)
b)
f(x)
x
2
x 1
f ‘(x)
INQUISITIVE MINDS want to know…
1.
If f ‘(a)= 0 does that necessarily mean f(x) changes from
increasing to decreasing (or vice versa) at x = a???
2.
If f(x) changes from increasing to decreasing (or vice versa) at
x = a does that necessarily mean there is a max/min at x = a???
What else could occur at x = a???
Ex 
Sketch a graph of a function, f(x), that is differentiable and that satisfies the
following conditions:
 f ’(x) < 0 when x > 4 and x < 0
 f ‘(x) > 0 when 0 < x < 4
 f(0) = –1
y
x
Ex 
For the cubic function defined by, f(x) = –2x3 + bx2 + cx + d, determine the
values of b, c, and d such that f(x) decreases to the point (0,2), increases to
the point (2,10) and then continues to decrease.
Ex 
Each of the following graphs represent the derivative function, f x  , of a
function, f x  . Determine the intervals of increase and decrease of f x 
and the x–coordinate for all local extrema of f x  . Then, sketch f x 
assuming f 0   0 .
5
a)
5
b)
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
1
2
3
4
5
-5 -4 -3 -2 -1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
5
5
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
1
2
3
4
5
-5 -4 -3 -2 -1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
2
3
3
4
4
5
5
Homework: p.169–171 #1–9, 11, 12