Search results
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
Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
May 22, 2024 · Solved Examples on Concavity. Example 1: Determine the intervals where the function f(x)=x 3 −6x 2 +9x+15 is concave up and concave down. First Derivative: f'(x)=3x 2-12x + 9. Second Derivative: f"(x)=6x-12. Find Critical Points for Concavity: Set the second derivative equal to zero to find potential points of inflection: 6x-12=0. x=12/6=2
In general, without having the graph of a function f, how can we determine its concavity? By definition, a function f is concave up if f′ is increasing. From Corollary 3, we know that if f′ is a differentiable function, then f′ is increasing if its derivative f′′(x) > 0.
- 784KB
- 7
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.
Definition 1 A function f defined on the convex set C ⊂ Rn is called con-cave if for every x1, x2 ∈ C and 0 ≤ t ≤ 1, we have. f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2). Definition 2 A function f defined on the convex set C ⊂ Rn is called strictly concave if for every x1 6= x2,and 0 < t < 1, we have. f(tx1 + (1 − t)x2) > tf(x1) + (1 − t)f(x2).
People also ask
What is the concavity of a function?
Why is concavity important in mathematics?
Why are concave functions important?
How do we generalize concave functions?
How do you know if a function is concave?
Is F a concave function?
Convexity is a term that pertains to both sets and functions. For functions, there are di erent degrees of convexity, and how convex a function is tells us a lot about its minima: do they exist, are they unique, how quickly can we nd them using optimization algorithms, etc.
