How derivatives affect the shape of a graph
(Section 3.3)
Alex Karassev
First and second derivatives


f ′ tells us about intervals of increase and
decrease
f ′′ tells us about concavity
First derivative:
Intervals of Increase / Decrease
Increasing / Decreasing
Slope of tangent line
0 f 
0 f 
slope > 0
y = f(x)
slope < 0
decr.
incr.
x
Increasing / Decreasing Test
Derivative
f ( x)  0 on an interval  f  on this interval
f ( x)  0 on an interval  f  on this interval
f ′ (x) > 0
y = f(x)
f ′ (x) < 0
decr.
incr.
x
Change of behavior
f can change from increasing to decreasing and vice versa:
• at the points of local max/min (i.e. at the critical numbers)
• at the points where f is undefined
f ′ (x) > 0
y = f(x)
f ′ (x) < 0
decr.
incr.
x
Local max/min: 1st derivative test
loc. max
loc. min.
f ′ (x) < 0
decr.
y = f(x)

Let c be
a critical number

How do we determine
whether it is loc. min or
loc. max or neither?
f ′ (x) > 0
c
incr.
x

If f ′ changes from negative to positive at c, it is loc. min.

If f ′ changes from positive to negative at c, it is loc. max.

If f ′ does not change sign at c, it is neither (e.g. f(x) = x3, c =0)
Second derivative:
Concavity
Concavity: definition
Graph lies above tangent lines:
concave upward
Graph lies below tangent lines:
concave downward
Concavity: example
Inflection points
up
down
y = f(x)
up
down
up
Concavity test: use f′′
f′′ (x) > 0
Graph lies above tangent lines:
concave upward
f′′ (x) < 0
Graph lies below tangent lines:
concave downward
Inflection points:
Numbers c where f′′(c) = 0 are "suspicious" points
Change of concavity
f can change from concave upward to
concave downward and vice versa:
• at inflection points (check f ′′ (x) = 0)
y = f(x)
• at the points where f is undefined
up
down
up
down
up
Local max/min: 2nd derivative test
loc. max
f ′′ (c) < 0
loc. min.
f ′′ (c) > 0
c
y = f(x)
x

Suppose f ′ (c) = 0

How do we determine
whether it is loc. min or
loc. max or neither?
NOTE:
tangent line at (c,f(c))
is horizontal

If f ′′ (c) > 0 the graph lies above the tangent ⇒ loc. min.

If f ′′ (c) < 0 the graph lies below the tangent ⇒ loc. max.

If f ′′ (c) = 0 the test is inconclusive (use 1st deriv. test instead!)
Comparison of 1st and 2nd derivative
tests for local max/min

Second derivative test is faster then 1st derivative test
(we need to determine where f′(c) = 0 and then just
compute f′′(c) at each such c)

Second derivative test can be generalized on the case
of functions of several variables

However, when f′′(c) = 0, the second derivative test is
inconclusive (for example, (0,0) is an inflection point
for f(x) = x3, while for x4 it is a point of local minimum,
and for –x4 it is a point of local maximum)
Examples


Sketch the graph of function y = x4 – 6x2
Use the second derivative test to find points
of local maximum and minimum of
f(x) = x/(x2+4)