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Economics 214

Economics 214. Lecture 7 Introduction to Functions Cont. Extreme Values. The largest value of a function over its entire range is call its Global maximum. The smallest value over the entire range is called the Global minimum. Largest value within a small interval is a local maximum.

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Economics 214

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  1. Economics 214 Lecture 7 Introduction to Functions Cont.

  2. Extreme Values • The largest value of a function over its entire range is call its Global maximum. • The smallest value over the entire range is called the Global minimum. • Largest value within a small interval is a local maximum. • Smallest value within a small interval is called a local minimum.

  3. Figure 2.8 A Function with Extreme Values

  4. Average Rate of Change

  5. Average Rate of Change Linear Function

  6. Average Rate of Change for Nonlinear function

  7. Secant Line

  8. Graph of Average Rate of Change for Nonlinear Function

  9. Concavity and Convexity • The concavity of a univariate function is reflect by the shape of it graph and its secant line. • Function is strictly concave in the interval [xA,xB] if the secant line drawn on that interval lies wholly below the function. • Function is strictly convex in the interval [xA,xB] if the secant line drawn on that interval lies wholly above the function.

  10. Figure 2.10 Strictly Concave and Strictly Convex Functions

  11. Strictly Concave

  12. Strictly Convex

  13. Convexity and secant line

  14. Our Example

  15. Concave

  16. Convex

  17. Figure 2.11 Concave and Convex Functions

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