If f is a concave f x 0 convex f x 0 function in an interval I then c is a

# If f is a concave f x 0 convex f x 0 function in an

• Notes
• 4

This preview shows page 3 - 4 out of 4 pages.

If f is a concave ( f ’’( x ) 0) / convex ( f ’’( x ) 0) function in an interval I then c is a maximum / minimum point for f in I . 8.2 Example Show that f is convex. Find its minimum point. x = 1 is a stationary point for the convex function f , so x = 1 minimizes f . 1 ( ) , ( ) x f x e x D f - = - = = ° 1 '( ) 1 x f x e - = - 1 ''( ) 0 so is convex x f x e f - = > = 1 '( ) 1 0 x f x e - = - = 1 0 1 x e e - = = = Lecture 6 8.1 Introduction *Extreme points: max and min *Necessary first-order condition *Stationary points 8.2 Simple Tests for Extreme Points T1*First derivative test for max and min T2*Max/min for concave/convex functions 8.3 Economic Examples *Examples for T1 and T2 Theorem 2 Theorem 2 : max/min for concave/convex functions Suppose that a function f is *C1: differentiable ( 2 times ) in an interval I *C2: c is an interior point of I and f’(c)=0 . If f is a concave ( f ’’( x ) 0) / convex ( f ’’( x ) 0) function in an interval I . then c is a maximum / minimum point for f in I . 8.3 Economic example theorem 2 The total cost of producing Q units of a good is: Find the value of Q which minimizes the average cost A ( Q ). So Q = 4 is the minimum point. 2 ( ) 2 10 32, Q 0 C Q Q Q = + + > + > ( ) 32 ( ) 2 10 C Q A Q Q Q Q = = + + = + 2 32 '( ) 2 A Q Q = - = 3 64 ''( ) 0, so is convex A Q A Q = > = '( ) 0 for 4 A Q Q = = = = 8.2 Simple tests for extreme points Theorem 1a : First derivative test for maximum . Suppose that a function f is *C1: differentiable in an interval I *C2: c is an interior point of I and f’(c)=0 If '( ) 0 for and '( ) 0 for then is a m axim um point for .

Subscribe to view the full document.

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes