Analyzing Convex and Concave Functions
Explore the properties of convex, concave, and linear functions through graphical representations and mathematical equations. Understand the relationship between concavity, convexity, and linearity in functions, illustrated with examples and diagrams. Learn how to identify and distinguish between these different function types.
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( ) 1 i pu r n m + + ( ) 1 i pu r a b a b i i = = i 1 pau r 1 i i n m ( ) ( ) pv r pv r i i i i i = = n n m m 1 1 + + i i ( ) ( ) ( ) ( ) p b pau r p b i i i i + = = = = 1 1 pb 1 1 i i i i i i n n m m + ( ) ( ) pau r pau r pb i i i i = = = = 1 1 1 1 i i i i
( ) E u for L EV(L) CE(L) RP(L)=EV(L)-CE(L)>0
( , ( )) x u x 2 2 ( , ( )) x u x ( , ( )) x u x 2 2 ( , ( )) x u x 2 2 1 1 ( , ( )) x u x ( , ( )) x u x 1 1 1 1 2x 1x 1x 2x 2x ) 1x + (0,1) (1 x x 1 2 Convex Function Linear Function Concave Function From the above figures it can seen that: 1. u(x) is a concave if this means that if you draw a line parallel with y axiom this line first will cut the curve of u(x) and then the cord between and . 2. u(x) is a convex if this means that if you draw a line parallel with y axiom this line first will cut the cord between and then the curve of u(x) . 3. u(x) is a linear function if + + ) ) x ( ) u x + ) ( ) u x [0,1], ( (1 (1 u x 1 2 1 2 ( , ( )) x u x ( , ( )) x u x 1 1 2 2 + ) ) x ( ) u x + ) ( ) u x [0,1], ( (1 (1 u x 1 2 1 2 ( , ( )) x u x ( , ( )) x u x 1 1 2 2 ) ) x = ( ) u x + ) ( ) u x [0,1], ( (1 (1 u x 1 2 1 2
= ( ( )) ( ) u CE L E u for L + + ( (1 ) , p x pu x ( ) (1 ) ( )) p u x px 1 2 1 2 ( ( ), ( ( )) CE L u CE L
= ( ) u x x = = = 1 2 ( ) u x x x = 1 2 1 2 1 ( ) u x x x 1 2 1 2
