Optimizing a Function (with active graph)

OPTIMIZING FUNCTIONS
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INDEX OF TERMS

Critical Value Types (i) / (ii) Graph Test Relative Maximum How to Optimize
Nearby Candidate 1st Derivative Test Relative Minimum 2nd order Critical Value
Ledge Inflection Point 2nd Derivative Test Concave up / down How to Test for Inflection Points

Try interactive SELF-TEST on these terms ?

Definition 1 A CRITICAL VALUE (CV) of a function f is a number c for which either:

f '(c) = 0 (CV of type (i))
or
f '(c) is undefined (CV of type (ii))

Note 1: CVs of type ( i ) are at C, E, G, and J below.

Note 2: CVs of type ( ii ) are A, F, K, and L below; A and L are domain endpoints.

Note 3: At F and K, tangents to the graph of f are VERTICAL, so f ' is undefined.

Note 4: If x=c is an endpoint in the domain of f (as A and L in figure below),
then we understand that f '(c) is UNDEFINED: a CV of type (ii) above.

Definition 2

Two numbers are NEARBY if NO CRITICAL VALUE lies between them.

Definition 3	A RELATIVE MAXIMUM (REL MAX) of f is a number x=a for which f(a) f(x) for all nearby numbers x. (A, E, and J in figure below).

Definition 4	A RELATIVE MINIMUM (REL MIN) of f is a number x=a for which f(a) f(x) for all nearby numbers x (A, C, F, and L in figure below).

Definition 5	A RELATIVE EXTREMUM of f is a number x=a which is either a relative maximum or a relative minimum of f.

Definition 6	A LEDGE VALUE (LV) of f, or simply LEDGE, is a number x=a which is a critical value but NOT a relative extremum. In the figure below, ledge values occur at G (horizontal ledge) and K (vertical ledge).

Move the mouse arrow slowly across the graph at the right. Various features of the graph will be pointed out.

Fact 7: [basic to optimizing] Every relative extremum is a CV.

Fact 8: Not all CV's are relative extrema (as G and K above).

Definition 9: A graph or function is called CONCAVE UP in an interval if the graph lies ABOVE it's tangents in that interval. Similarly, this graph is called CONCAVE DOWN if it lies BELOW it's tangents. Concavity is illustrated in the graph shown above

Fact 10a: If f "(x) > 0 in an interval, then f is concave up there.
(e.g., use a mouse on intervals [B,C] or [C,D] in the figure above)

Fact 10b: If f "(x) < 0 in an interval, then f is concave down there.
(e.g., use a mouse on intervals [E,F] or [F,G] in the figure above)

Definition 11: A CANDIDATE of f (or 2nd order Critical Value) is a number x = k at which either f "(k) = 0 or f "(k) is undefined. It is not always possible to distinguish candidates by the shape of a graph, but points A, B, D, F, H, K and L are candidates above.

Definition 12: An INFLECTION POINT (IP) is a candidate at which f "(x) changes sign, and hence (see Facts 10a/10b) concavity changes. Points B, D, G, and K above are IPs, as well as candidates.

OUTLINE OF PROCEDURE FOR OPTIMIZING f(x)

I.	Find f '(x)
II.	Find all CV's of f(x)
III.	Test each CV as to what kind of extremum (if any) it is.

There will be 3 tests available, as follows:

IIIA. Graph test (stated below)
IIIB. 1^st derivative test (see text Pg.172 and below)
IIIC. 2^nd derivative test (see text Pg.189 and below)

III A. Graph test for CV: x = c

Calculate f(x) at x = c and at two nearby test values, x = b and x = d,
with b

d: see figure at right.

If BOTH f(c)

f(b) AND f(c)

f(d), then c is a relative maximum,
as in GRAPH A at the right.

If BOTH f(c)

f(b) AND f(c)

f(d), then c is a relative minimum,
as in GRAPH B at the right.

Otherwise, c is a ledge (LV), as in GRAPH C at the right

III B. 1^st derivative test for CV: x = c

Calculate f '(x) at 2 nearby test values, x = b and x = d, with b

d: see figure above.

If f '(b)

O , and f '(d)

O , then x = c is a relative maximum; again, see GRAPH A above.

If f '(b)

O , and f '(d)

O , then x = c is a relative minimum; again, see GRAPH B above.

Otherwise, x = c is a ledge; again, see GRAPH C above.

III C. 2^nd derivative test for CV of type (i): x = c

Calculate f "(x) at x = c.

If f "(c) < 0 , then x = c is a relative maximum; again, see GRAPH A above.

If f "(c) > 0 , then x = c is a relative minimum; again, see GRAPH B above.

If f "(c) = 0 , then x = c could be a max, min, or ledge: use IIIA or IIIB to see which.

PROCEDURE FOR ANALYZING IPs
(inflection points)

IV. Find f "(x)

V. Find all candidates (see Definition 9 above).

VI. Test each candidate k to see whether or not it is an inflection point, as follows:

VIa.	Calculate f "(b) and f "(d), where bkd; both b and d must be nearer to k than the next nearest candidate.
VIb.	Only if f "(b) and f "(d) have opposite signs will x = k be an IP.

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SELF-TEST
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This page last updated
12 October 2011

Definition 9:	A graph or function is called CONCAVE UP in an interval if the graph lies ABOVE it's tangents in that interval. Similarly, this graph is called CONCAVE DOWN if it lies BELOW it's tangents. Concavity is illustrated in the graph shown above

Definition 11:	A CANDIDATE of f (or 2nd order Critical Value) is a number x = k at which either f "(k) = 0 or f "(k) is undefined. It is not always possible to distinguish candidates by the shape of a graph, but points A, B, D, F, H, K and L are candidates above.

Definition 12:	An INFLECTION POINT (IP) is a candidate at which f "(x) changes sign, and hence (see Facts 10a/10b) concavity changes. Points B, D, G, and K above are IPs, as well as candidates.