Concavity and the Second
Derivative Test
Calculus 3.4
Concavity
• f is differentiable on an open interval
• The graph is concave upward if f´ is
increasing on the interval.
• Graph lies above its tangent lines
• Graph “holds water”
• The graph is concave downward if f´ is
decreasing on the interval.
• Graph lies below its tangent lines
• Graph doesn’t “hold water”
Calculus 3.4
2
Finding Concavity
• If f´´ > 0; the graph of f is concave up.
• f´ is increasing
• If f´´ < 0; the graph of f is concave
down.
• f´ is decreasing
• If f´´ = 0, the graph of f is linear, and
neither concave up nor concave down.
• then f´ is constant
Calculus 3.4
3
Points of inflection
• The concavity of f changes
• The graph of f crosses its tangent line.
• May occur when f´´ is 0 or undefined.
Calculus 3.4
4
Examples
• Determine the points of inflection and
discuss the concavity of the graph of the
function.
2
f  x   x  3x  8
f  x   x3  9x2  27 x
Calculus 3.4
5
The Second Derivative Test
• If f´(c) = 0 and f´´(c) > 0, then f(c) is a
relative minimum.
• If f´(c) = 0 and f´´(c) < 0, then f(c) is a
relative maximum.
• If f´(c) = 0 and f´´(c) = 0, then the test
fails.
• Use the first derivative test
Calculus 3.4
6
Examples
• Determine the points of inflection and
relative extrema for the function. Use
the second derivative test if possible.
3
f  x  x 1
f  x  x x 1
f  x   2sin x  cos 2 x
0  x  2Calculus
 3.4
7
Examples
• Determine the relative extrema for the
function. Use the second derivative test
if possible.
f  x   2sin x  cos 2 x
0  x  2
Calculus 3.4
8
Example
• Sketch the graph of a function f having
the following characteristics
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
y
4
f  0  f  2  0
3
f   x   0 if x  1
2
1
f  1  0
f   x   0 if x  1
-5
-4
-3
-2
-1
x
1
2
3
4
5
-1
f   x   0
-2
-3
-4
Calculus 3.4
-5
9
Example
• The graph of f is
shown. On the
same set of axes,
sketch the graphs
of f´ and f´´.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
y
4
3
2
1
-5
-4
-3
-2
-1
x
1
2
3
4
5
-1
-2
-3
-4
-5
Calculus 3.4
10