3.5 PROBLEMS In Problems I through 10, state whether the given function at tains a maximum value or a minimum value (or both) on the given interval (Suggestion: Begin by sketching a graph of the function/ 1. f(x) 1-x; -1.1) $f(x)=x: (1.1) 5. f(x)=x-21: (1.4) 7. f(x)+1; -1.1] & f(x)= (-00,00) 1 9. f(x) = x(1-x) 10. f(x) = 1. ZOTO this lo S=0.4 Calcula factor like a horizontal line (as in Fig. 3.5.1 in this manner? ke the c zontal ta Ordinate of esponding point with m alue attainet 1 2 f(x)=2x+1: (1.1) 4. 700) = (0.1) 6. f(x)=5-²: 1-1,2) [2,3] (0,1) *(1-x) In Problems 11 through 40, find the maximum and minimum values attained by the given function on the indicated closed interval 11. f(x) = 3x-2; [-2.3] 12 f(x)=4-3x: [-1.5] 13, h(x)=4-x²¹: [1.3] 14. f(x)=x²+3; [0.5] 1 g(x)=(x-1): [-1.4] 16. h(x)=x² + 4x +7: [-3.0] 17. f(x)=x²-3x: [-2,4] 18 g(x)=2r³-9x² + 12x: (0.4] 19. h(x)=x+ [1.4] 16 [1.3] 21. f(x)=3-2x; [-1.1] 22. f(x)=x²-4x+3; [0,2] 23. f(x)=5-12x-9x²; [-1,1] 24. f(x)=2x² - 4x +7; [0.2] 25. f(x)=x²-3x²-9x+5; [-2.4] ed interval (a. 20. f(x)=x² + = (if any) of f. minimum nor a ka ossibility that fis ferentiable there xtremum in (a, b) m, but only one ca local minimum, and in (a.b). 28. f(x)-(2x-3): [2.2] 29. fux)-5+17-3x: (1.5) 30. f(x)=x+11+x-11: (-2,23 31. f(x)=50-105x²+72x: (0.1) 32. fux)=2x+ (1.4) 33, 70x)= (0.3) 26. f(x)=r³+x: [-1.2] 27. f(x)=3x³-5x³; [-2,2] *+1 34. f(x)=110,3) 1-x 35. f(x)= 56. fox)-2- -1.8) 37. f(x)=x√1-¹: [-1.1] 38. f(x)=x√/4-¹ (0.21 39. f(x)=x2-x) (1.3) 40. f(x)=x2-x² (0.4) 41. Suppose that fix) Ax+ B is a linear function and that A 0. Explain why the maximum and minimum values of fon a closed interval (a, b) must occur at the endpoints of the interval. (-2,5) 42. Suppose that is continuous on [a, b] and differentiable on (a, b) and that f'(x) is never zero at any point of (a, b). Explain why the maximum and minimum values of f must occur at the endpoints of the interval (a, b). 43. Explain why every real number is a critical point of the greatest integer function f(x)=[x]. 44. Prove that every quadratic function f(x) = ax²+bx+c (0) has exactly one critical point on the real line. 45. Explain why the cubic polynomial function f(x) = ax + bx² +cx+d (a*0) can have either two, one, or no critical points on the real line. Produce examples that illustrate each of the three cases. 46. Define f(x) to be the distance from x to the nearest integ What are the critical points of f