Proof of functional concave and convex by using quadratic Derivative

Source: Internet
Author: User

Many people actually know that the secondary derivative of the function can be used to judge the concave and convex of the function, but many people forget how to prove it. Here I will prove it again.

Proof: If f (x) is continuous in (A, B) and can be imported twice, if f'' (x)> 0, the function is concave. If f (x) is greater than 0, the function is convex.


Preface:

First, we will give several theorems and explanations.

Description of Function Concave and convex:

Function f (x) is continuous in (a, B). For any A <X1 <X2 <B, X0 = (X1 + x2)/2,

If F (x0) <(f (X1) + f (X2)/2, the function (up) is considered concave; if F (x0)> (f (X1) + f (X2)/2, the function (up) is considered convex.


A Theorem to be used in the process of proof is given: The mean value theorem of Laplace.

If function f (x) is continuous in [a, B] In (a, B), at least one e exists, and a <e <B causes f' (E) = (F (B)-f (a)/(B-)


The following describes our initial problems:

Any f (x) on two points x1, x2 makes a <X1 <X2 <B, X0 = (X2 + X1)/2, then the x0-x1 = x2-x0, let the x0-x1 = x2-x0 = H, then apply two Laplace Mean Value Theorem in (x1, X1 + H) and (x2-h, X2) respectively

F' (E1) = (f (x0)-f (X1)/H (1)

F' (E2) = (f (X2)-f (x0)/H (2)

E1 is in the range of (x1, X0) and E2 is in the range of (x0, X2 ).

Let (1) and (2) Multiply H to the left, and then subtract to (f' (E2)-f' (E1) H = f (X2) + f (X1)-2 (f (x0) (3)

Then, we use the mean value theorem of a Laplace in (E1, E2) to obtain the following result:

F' (E2)-f' (E1) = f'' (E) (e2-e1) Where E is within the range of (E1, E2, then, if the value of F ''(e) is greater than 0, the left side of (3) is greater than 0. Likewise, the right side of (3) can be obtained.

F (x0) = f (X1 + x2)/2) <(f (X2) + f (X1)/2, combined with the convex theorem of the function, we can obtain that the original function is concave, And the convex proof is similar .. No proof.

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.