# Math 112 Solutions for 3.7, p. 286

2.
describes a parabola opening upward with zeros at 0 and and vertex . has the same zeros but vertex and opens downward.
3.
Let . The critical points are 0 and . and . Hence if and , has global minima at and a local maximum at 0. If and , has only a global minimum at 0. If and , has global maxima at and a local minimum at 0. If and , has only a global maximum at 0.

4.
Let . The critical points are at and the inflection point is at . The average of the -coordinates of the critical points is clearly the -coordinate of the inflection point. The same is true for the -coordinates.
5.
If the cubic in Problem 4 has only one critical point, it is because and the critical point is , the inflection point.
8.
Let the cubic have the equation . If (3, 0) is the local maximum and (5, -1) is the inflection point, then the local minimum is at (7, -2) and is positive. The inflection point is at , so . Since the first critical point is , we get . Finally, using , we get . Hence with .

10.
(a) Since 120 mi/hr = 176 ft/sec, the maximum steepness is , so the minimum slope is .

(b) With , we let the inflection point be the origin; since the inflection point is at , we have . Since , we have . The slope at the inflection point is . The critical points are then at and . Therefore the run is and the rise is .

(c) If the rise is -12,000 feet, then the run is feet miles.

(d) From part (c) we have .

11.
Increasing magnifies the graph vertically. Increasing compresses the graph horizontally.

13.
If the maximum of is (7,55), then and . It follows that is less than 20 when , which is true for .

14.
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17.
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20.
Demand will be about 24 sales per week, and it will take another four weeks to reach it.

22.

25.
Think of as an amplitude to a sunisoid.

26.
The graph is a bell curve having its maximum at and inflection points at .

27.
Let the chain hang from the points and with its low point at the origin. Since the low point is a critical point, we get , so . Since the origin is on the curve, we have , and . Then , so . Thus .