Guía 12: Optimización II. Concavidad y convexidad. Exámenes: Capítulos 9.5, 6.1. - Prof. 3, Apuntes de Negocios Internacionales

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Guide 12. Block 3: Optimization II.

Concavity and convexity.

S&H Chapters 9.5, 6.

Recommended problems: 9.5: 1–9. 6.1: 5.

Index

Optimization II (a) The second derivative of a function (b) Concavity and convexity (c) Inflection points

Graphic representation of functions

Optimization II (b) Concavity and convexity

Definition of concavity and convexity for functions twice

differentiable

Given a function that is continuous in the interval [a, b] and twice differentiable in the interval (a, b), we say that (S&H p. 305):

1. f (x) is convex in (a, b) ⇔ f ′′(x) ≥ 0 for all x ∈ (a, b) ⇔ f (x) has the shape “

2. f (x) is concave on (a, b) ⇔ f ′′(x) ≤ 0 for all x ∈ (a, b) ⇔ f (x) has the shape “

Attention: sometimes the words “concave” and “convex” are used with different meanings in some books. We should check the definition in every book.

Optimization II (b) Concavity and convexity

Growth and concavity

Combining the possible growth and concavity of a function, we have 4 possible models: f (x) convex (f ′′(x) ≥ 0) “

f (x) concave (f ′′(x) ≤ 0) “

f (x) increasing (f ′(x) ≥ 0) 0 0

f (x) decreasing (f ′(x) ≤ 0) 0 0

Optimization II (c) Inflection points

Caracterization of inflection points

Theorem 1 (S&H Theorem 9.3)

If f is a function with a continuous second derivative in (a, b) and c ∈ (a, b), then

1. If c is an inflection point, then f ′′(c) = 0.
2. If f ′′(c) = 0 and f ′′^ changes sign at c, then c is an inflection point of f.

Optimization II (c) Inflection points

The graph of the inflection points

Optimization II (c) Inflection points

Summarizing:

f (x) =

x^3 −

x^2 −

x + 1

x − 1 1 / 2 2 Sign f ′(x) + 0 − 0 + Sign f ′′(x) − 0 + f (x) ↗ max ↘ min ↗ shape

i.p.

Graphic representation of functions

Graphic representation of functions

Summary of the procedure:

1. Domain. Points of discontinuity and vertical asymptotes.
2. Intersection points with the x axis (f (x) = 0).
3. Sign of f (x) (recommended).
4. Intersection points with the y axis.
5. Stationary points.
6. Local maximum and minimum points.
7. Concavity, convexity and inflection points.
8. Horizontal asymptotes. Recall: I (^) x = a is a vertical asymptote if limx→a f (x) = ±∞. I (^) y = A is an horizontal asymptote of f (x) at +∞ if limx→+∞ f (x) = A. I (^) y = B is an horizontal asymptote of f (x) at −∞ if limx→−∞ f (x) = B.