# Section 4.2: Second Derivatives and Graphs

## Concavity

• 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).

## Second Derivative

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:

d2y/dx2 and y''

### Second Derivative and Concavity

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

### Inflection Points

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

### Locating Inflection Points

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

### Summary: Graphing f(x)

 +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?

c = -4

## Graphing Strategy (First Version)

1. 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).
2. 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.
3. 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.
4. 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.