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    gt gt Presentation Transcript

    • Graph Theory Problems
    • Problem #1. Find the local extrema, the point of inflection, and the intervals on which the function is increasing, decreasing, concave up, and concave down. F(x) = -3x 3 +8x 2 +4
      • F(x) = -3x 3 +8x 2 +4
      • F’(x) = -9x 2 +16x =
      • x(-9x+16)
      • F’(x) = 0 when
      • X = 0 or -9x+16 = 0
      • So F’(x) = 0 at x=0 and x=16/9
      • The function has a local minimum at x=0 since f changes from decreasing to increasing.
      • The function has a local maximum at x=16/9 since f changes from increasing to decreasing.
      • The function is increasing on the interval (0,16/9) since F’(x) is positive on that interval.
      • F’(x) = -9x 2 +16x = x(-9x+16)
      • F’’(x) = -18x+16
      • F’’(x) = 0 when
      • X = 8/9.
    •  
      • 2. Use the graph of f’(x) to estimate where the function has the local extrema, the point of inflection, and the intervals on which the function is increasing, decreasing, concave up, and concave down.
    •  
    •  
    • TRY THE NEXT TWO PROBLEMS…!
      • (PROBLEM #3 AND #4 DON’T HAVE SOLUTIONS.)
    • Problem #3) Find the local extrema, the point of inflection, and the intervals on which the function is increasing, decreasing, concave up, and concave down.                                    
      • ANSWER:
    • Problem#4) The graph below is f’’(x). Find the point of inflection of f(x) and the intervals on which the function concaves up and concaves down.
      • ANSWER: