- The graph of a function
*f*is**concave upward**on the interval (*a, b*) if*f'(x)*is*increasing*on (*a, b*). - The graph of a function
*f*is**concave downward**on the interval (*a, b*) if*f'(x)*is*decreasing*on (*a, b*).

For *y = f(x)*, the **second derivative** of *f*, provided that it exists, is:

f''(x) = d/dx f'(x)

Other notations for *f''(x)* are:

dand^{2}y/dx^{2}y''

For the interval (*a, b*), if *f'' > 0*, then *f* is concave upward, and if *f'' < 0*, then *f* is concave downward.

An **inflection point** is a point on the graph where the concavity changes from upward to downward or downward to upward.

If (*c, f(c)*) is an inflection point of *f*, then *c* is a partition number for *f''*.

+ve | -ve | 0 | Not defined | |
---|---|---|---|---|

f(x) | Above x-axis | Below x-axis | x-intercept | Vertical asymptote? |

f'(x) | Increasing | Decreasing | Local extremum? | Local extremum? |

f''(x) | Concave upwards | Concave downwards | Inflection point? | Inflection point? |

*Analyze f(x)*.- Find the domain and the intercepts.
- The
*x*intercepts are the solutions of*f(x) = 0*. - The
*y*intercept is*f(0)*.

*Analyze f'(x)*.- Find the partition numbers for
*f'*and the critical numbers of*f*. - Construct a sign chart for
*f'(x)*. - Determine the intervals on which
*f*is increasing and decreasing. - Find the local maxima and minima of
*f*.

- Find the partition numbers for
*Analyze f''(x)*.- Find the partition numbers for
*f''*. - Construct a sign chart for
*f''(x)*. - Determine the intervals on which the graph of
*f*is concave upward and concave downward. - Find the inflection points of
*f*.

- Find the partition numbers for
*Sketch the graph of f*.- Locate intercepts, local maxima and minima, and inflection points.
- Sketch in what is known from steps 1 - 3.
- Plot additional points as needed and complete the sketch.