Math 140 - CooleyBusiness Calculus OCC
Section 2.2 –Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Definition
Suppose that f is a function whose derivative f exists at every point in an open interval I. Then
f is concave up on I if f is increasing over I.
f is concave down on I if f is decreasing over I.
Theorem 4 – A Test for Concavity
- If f(x) > 0 on an interval I, then the graph of f is concave up.
( f is increasing, so f is turning up on I.)
- If f(x) < 0 on an interval I, then the graph of f is concave down.
( f is decreasing, so f is turning down on I.)
Theorem 5 – The Second-Derivative Test for Relative Extrema
Suppose that f is differentiable for every x in an open interval (a, b) and that there is a critical value c in
(a, b) for which f(c) = 0. Then:
- f(c) is a relative minimum if f(c) > 0. (f is concave up)
- f(c) is a relative maximum if f(c) < 0. (f is concave down)
For f(c) = 0, the First-Derivative test can be used to determine whether f(x) is a relative extremum.
Definition
An inflection point (point of inflection) for a function f, is the point across which the direction of concavity changes.
Theorem 6 – Finding Points of Inflection
If a function f has a point of inflection, it must occur at a point x0, wheref(x0) = 0 or f(x0) does not exist.
Strategy for Sketching Graphs – (later to be expanded on and refined in Section 2.3)
a.)Derivatives and domain. Find f(x) and f(x). Note the domain of f.
b.)Critical values of f. Find the critical values by solving f(x) = 0 and finding where f(x) does not exist. These numbers yield candidates for relative maxima and minima. Find the function values at these points.
c.)Increasing and/or decreasing: relative extrema. Substitute each critical value, x0, from (b) into f(x). If f(x0) < 0, then f(x0) is a relative maximum and f is increasing to the left of x0 and decreasing to the right. If f(x0) > 0, then f(x0) is a relative minimum and f is decreasing to the left of x0 and increasing to the right.
d.)Inflection points.Determine candidates for inflection points by finding where f(x) = 0 or where f(x) does not exist. Find the function values at these points.
e.)Concavity. Use the candidates for inflection points from step (d) to define intervals. Substitute test values into f(x) to determine where the graph is concave up (f(x) > 0) and where it is concave down (f(x) < 0).
f.)Sketch the graph. Sketch the graph using the information from steps (a) – (e), calculating and plotting extra points as needed.
Exercises:
Sketch the graph of each function. List the coordinatesof where extrema or points of inflection occur. State
where the function is increasing or decreasing, as well as where it is concave up or down.
1)
2)
Exercises:
Sketch the graph of each function. List the coordinatesof where extrema or points of inflection occur. State
where the function is increasing or decreasing, as well as where it is concave up or down.
3)
4)
1