Download Section 5.1

AP CALCULUS NOTES
SECTION 5.1 ANALYSIS OF FUNCTIONS I: INCREASE, DECREASE, AND
CONCAVITY
The terms increasing, decreasing, and constant are used to describe the behavior of a function
over an interval as we examine a graph from left to right.
A.) Increasing and Decreasing Functions – Let f be defined on an interval, and let x1 and x2
denote numbers in that interval.
 f is increasing on the interval if f  x1   f  x2  whenever x1  x2 .  f  x2   f  x1   0

f is decreasing on the interval if f  x1   f  x2  whenever x1  x2 .  f  x2   f  x1   0

f is constant on the interval if f  x1   f  x2  for all x1 and x2 .  f  x2   f  x1   0
Note: f is increasing on any interval where its graph has positive slope, is decreasing on any
interval where its graph has negative slope, and is constant on any interval where its graph has 0
slope.
1.) Theorem – Let f be a function that is continuous on a closed interval  a, b and
differentiable on the open interval  a, b  .

If f   x   0 for every value of x in  a, b  , then f is increasing on  a, b  .

If f   x   0 for every value of x in  a, b  , then f is decreasing on  a, b  .

If f   x   0 for every value of x in  a, b  , then f is constant on  a, b  .
Ex.) Find the intervals where the function f  x   3x4  4x3 12x2  5 is increasing and
decreasing.
Critical Number of a function f – is a number c in the domain of f such that either f   c   0 or
f   c  does not exist.
Note: The sign of the derivative of f does not reveal the direction of curvature of the graph of f.
For this we need the 2nd derivative:
B.) Concavity – describes the direction of curvature of the graph of f.
The graph is concave up on intervals where the tangent lines have
increasing slopes and concave down on intervals where the tangent
lines have decreasing slopes.
Thus, if f is differentiable on an open interval I, then
 f is concave up on I if f  is increasing on I.
 f is concave down on I if f  is decreasing on I.
1.) Theorem – Let f be twice differentiable on an open interval I.
 If f   x   0 on I, then f is concave up on I.

If f   x   0 on I, then f is concave down on I.
9.)
Interval
 ,1
1, 2
 2,3
3, 4
 4,  
Sign of
f  x
Sign of
f   x 
-
+
+
+
+
-
-
-
-
+
C.) Inflection Points – A point P on a curve y  f  x  is called an inflection point if f is
continuous there and the curve changes the direction of its concavity at P.
21.) f  x   x1 3  x  4
Ex.) Sketch a possible graph of a function f that satisfies the following conditions:
1. f   x   0 on  ,1 , f   x   0 on 1,  
2.
3.
f   x   0 on  , 2 and  2,   , f   x   0 on  2, 2 
5
4
lim f  x   2, lim f  x   0
x
3
x 
2
1
-5 -4 -3 -2 -1
1
-1
-2
-3
-4
-5
2
3
4
5
Note: Inflection points mark the places on the curve y  f  x  where f   x  (the rate of change)
changes from increasing to decreasing, or vice versa.
Use the figure above to help you with determining the inflection points in #58.
58.)
42.b.) Show that e x  1  x 
1 2
x if x  0 .
2