745 B
Executable file
745 B
Executable file
Concavity and inflections
Concave up
A function f
is concave up on an open differentiable interval I
if the derivative f'
is an increasing function on I
, then f'' > 0
. Obtaining tangent line above the graph.
Concave dowm
A function f
is concave down on an open and differentiable interval I
if the derivative is a decreasing function on I
, then f'' < 0
. Obtaining tangent lines below the graph.
Inflection points
The function f
has an inflection point at x_0
if
- the tangent line in
(x_0, f(x_0))
exists, and - the concavity of
f
is opposite on opposite sides ofx_0
.
If f
has an inflection point at x_0
and f''(x_0)
exists, then f''(x_0) = 0
The second derivative test
...