Section 4.2: Second Derivatives and Graphs


Concave upwards

Concave downwards

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)

f(x)Above x-axisBelow x-axisx-intercept
f'(x)IncreasingDecreasingLocal extremum
f''(x)Concave upwardsConcave downwardsInflection point

f'(x) < 0, f(x) increasing

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.