Search results
This is just a quick and condensed note on the basic definitions and characterizations of concave, convex, quasiconcave and (to some extent) quasiconvex functions, with some examples. Contents. Concave and convex functions 1. 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Given the graph of a function, the hypograph of f,
- 227KB
- 12
May 22, 2024 · Concavity provides valuable insights into how a function curves, distinguishing between concave upward and concave downward shapes, while points of inflection mark locations where the curvature changes sign.
This notion is called the concavity of the function. Figure 4.34(a) shows a function f with a graph that curves upward. As x increases, the slope of the tangent line increases. Thus, since the derivative increases as x increases, f′ is an increasing function. We say this function f is concave up.
- 784KB
- 7
- a r i e s t h e angle , # t h e second v a r i e s t h e c i r c l e s
- a r i e s t h e angle , # t h e second v a r i e s t h e c i r c l e s
- a r i e s t h e angle , # t h e second v a r i e s t h e c i r c l e s
- a r i e s t h e angle , # t h e second v a r i e s t h e c i r c l e s
line = x [ na , : ] + r [ na , : ] np . s i n ( ∗ h i [ : , na ] ) line = y [ na , : ] + r [ na , : ] np . cos ( h i [ : , na ] ∗ ) ax= p l t . p l o t ( x line , y line , ’− ’ )
line = x [ na , : ] + r [ na , : ] ∗ np . s i n ( h i [ : , na ] ) line = y [ na , : ] + r [ na , : ] np . cos ( h i [ : , na ] ) ∗ ax= p l t . p l o t ( x line , y line , ’− ’ )
line = x [ na , : ] + r [ na , : ] ∗ np . s i n ( h i [ : , na ] ) line = y [ na , : ] + r [ na , : ] np . cos ( h i [ : , na ] ) ∗ ax= p l t . p l o t ( x line , y line , ’− ’ )
line = x [ na , : ] + r [ na , : ] np . s i n ( h i [ : , na ] ) ∗ line = y [ na , : ] + r [ na , : ] np . cos ( ∗ h i [ : , na ] ) ax= p l t . p l o t ( x line , y line , ’− ’ )
A simple (but non-obvious) computational geometry algorithm for determining convexity is to check that every triplet of consecutive vertices is CCW. This takes just a few lines of Python geometry code (assuming that the points are provided in counterclockwise order—if points is in clockwise order, you’ll want all triplets to be clockwise):
People also ask
What is the concavity of a function?
Why is concavity important?
How do you know if a function is concave?
Is f g a concave function?
Does F change concavity at a point?
How do you identify concavity and points of inflection?
Example Let f(x) = x3. Then f00(x) = 6x, so f00(x) < 0 for all x in (1 ;0) and f00(x) > 0 for all x in (0;1). Thus the graph of f is concave downward on (1 ;0) and concave upward on (0;1). We call the point (0;0) where the concavity of the graph changes a point of in ection.
