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1. What is an inflection point and how do you identify it? 2. How do you test a function to be convex or concave?

1. What is an inflection point and how do you identify it? 2. How do you test a function to be convex or concave?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.2: Trigonometric Equations
Problem 83E
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methods? Why or why not? EXERCISES 1. What is an inflection point and how do you identify it? 2. How do you test a function to be convex or concave? 3. What is the unimodal property and what is its significance in single-variable optimization? 4. Suppose a point satisfies sufficiency conditions for a local minimum. How do you establish that it is a global minimum? 5. Cite a condition under which a search method based on polynomial interpolation may fail. 6. Are region elimination methods as a class more efficient than point estimation may fail Or why 7. In terminating search methods, it is recommended that both the difference in variable values and the difference in the function values be tested. Is it possible for one test alone to indicate convergence to a minimum while the point reached is really not a minimum? Illustrate graphically. 8. Given the following functions of one variable: (a) fx) - x +x* -+2 (b) fix) = (2x + lyx - 4) Determine, for each of the above functions, the following: (i) Region(s) where the function is increasing; decreasing (ii) Inflexion points, if any (iii) Region(s) where the function is concave; convex (iv) Local and global maxima, if any (v) Local and global minima, if any 9. State whether each of the following functions is convex, concave, or neither. (a) f)-e (b) f) -e (c) fix) (d) f(x) - x+ log x (e) fix)- lel for x>0 (D f(x) - x log x (g) fx) - x (h) fix) for x>0 where k is an integer where k is an integer 10. Consider the function f(x) = x' - 12r + 3 over the region -4 x<4 Determine the local minima, local maxima, global minimum, and global maximum of f over the given region. 11. Carry out a single-variable search to minimize the function - 5 fx) - 3x + on the interval <x< using (a) golden section, (b) interval halving, Each search method is to use four functional evaluations only. Compare the final search intervals obtained by the above methods 12. Determine the minimum of f(x) = (10x + 3x +x + 5) starting at x = 3 and using a step size A= 5.0. Using region elimination: expanding pattem bounding plus six steps of golden section. II
Transcribed Image Text:methods? Why or why not? EXERCISES 1. What is an inflection point and how do you identify it? 2. How do you test a function to be convex or concave? 3. What is the unimodal property and what is its significance in single-variable optimization? 4. Suppose a point satisfies sufficiency conditions for a local minimum. How do you establish that it is a global minimum? 5. Cite a condition under which a search method based on polynomial interpolation may fail. 6. Are region elimination methods as a class more efficient than point estimation may fail Or why 7. In terminating search methods, it is recommended that both the difference in variable values and the difference in the function values be tested. Is it possible for one test alone to indicate convergence to a minimum while the point reached is really not a minimum? Illustrate graphically. 8. Given the following functions of one variable: (a) fx) - x +x* -+2 (b) fix) = (2x + lyx - 4) Determine, for each of the above functions, the following: (i) Region(s) where the function is increasing; decreasing (ii) Inflexion points, if any (iii) Region(s) where the function is concave; convex (iv) Local and global maxima, if any (v) Local and global minima, if any 9. State whether each of the following functions is convex, concave, or neither. (a) f)-e (b) f) -e (c) fix) (d) f(x) - x+ log x (e) fix)- lel for x>0 (D f(x) - x log x (g) fx) - x (h) fix) for x>0 where k is an integer where k is an integer 10. Consider the function f(x) = x' - 12r + 3 over the region -4 x<4 Determine the local minima, local maxima, global minimum, and global maximum of f over the given region. 11. Carry out a single-variable search to minimize the function - 5 fx) - 3x + on the interval <x< using (a) golden section, (b) interval halving, Each search method is to use four functional evaluations only. Compare the final search intervals obtained by the above methods 12. Determine the minimum of f(x) = (10x + 3x +x + 5) starting at x = 3 and using a step size A= 5.0. Using region elimination: expanding pattem bounding plus six steps of golden section. II
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