Quantitative Methods
Lecture 8
Sketching the graph of a function
•BAKVM

Outline of the lecture
•Sketching the graph of a function
(functions increasing and decreasing, convex and concave,
 local minima and maxima, inflection points)

Sketching the graph of a function



Intervals of monotonicity and of convexity and concavity
•Function increasing, non-decreasing, non-increasing, and decreasing
•Convex combination of points
•Function strictly convex, convex, concave, and strictly concave
•Characterization by the derivatives of the first, second and higher order

Sketching the graph of a function



Sketching the graph of a function



Sketching the graph of a function



Theorem



Theorem



Theorem



Theorem



Sketching the graph of a function



Sketching the graph of a function



Theorem



Theorem



Theorem



Theorem



Local extrema and inflection points
•Function increasing / decreasing
at a point
•The point of a
(strict) local maximum / minimum
•The line tangent to the graph
of a function at a point
•Function strictly convex / concave
at a point
•The point of inflection

Function increasing and decreasing at a point



Function increasing and decreasing at a point



Function increasing and decreasing at a point



Local extrema of a function



Local extrema of a function
increasing
↓
decreasing

Local extrema of a function
decreasing
↓
increasing

Local extrema of a function



Local extrema of a function



The line tangent to the graph of a function



Function strictly convex / concave at a point



Function strictly convex / concave at a point



Function strictly convex / concave at a point



Points of inflexion
convex
↓
concave

Points of inflexion
concave
↓
convex

Points of inflexion



Examples



Examples



Theorem



Examples



Examples