Four smooth curves—A, B, C, and D—are drawn on the same coordinate plane. Exactly three of the curves belong to the same family in the sense that one of them can be viewed as an original function, another as its first derivative, and the third as its second derivative. One remaining curve is unrelated to the other three.
Descriptions of the curves:
Curve A
• Crosses the x–axis at x=−1, 0, and 2.
• Decreasing on (−∞,−1) and (0,2); increasing on (−1,0) and (2,∞).
• Has a point of inflection at x=1.
Curve B
• Positive on (−1,2) and negative outside that interval.
• Zeros at x=−1 and x=2.
• Local maximum at x=0.
Curve C
• Always increasing; C(x)<0 for x<1 and C(x)>0 for x>1.
• Has a single zero at x=1.
Curve D
• Oscillatory with infinitely many zeros; bounded between y=−1 and y=1.
• No interval on which the graph is strictly increasing or strictly decreasing for more than one unit of x.
Use the descriptions to answer the questions below.
(1) Part 1: (a) Identify which three of the four curves form the related family and name the curve that is the distractor. Provide brief justification.
(2) Part 2: (b) For the three related curves, designate which is the original function, which is its first derivative, and which is its second derivative. State and justify your reasoning, referencing specific features from the descriptions.
(3) Part 3: (c) For the original function in your identification, determine the x–intervals where the function is concave down and decreasing simultaneously. Support your answer thoroughly.