Curve Sketching using Data Points

[mathnerd]

Hey guys, this is the question I am trying to solve:

The following five points lie on a function: (1, 20), (2, 4), (5, 3), (6, 2), and (10, 1).

Find an equation which passes through these points, and which also has the following:
. . .i) three inflection points
. . .ii) at least one local maximum
. . .iii) at least one local minimum
. . .iv) at least one critical point which differs from the listed points (above)
. . .v) the property that it is continuous and differentiable throughout
. . .vi) a piecewise definition (that is, it is not a single polynomial)

Now I understand that the graph should have 3 inflection points, one local maximum and one local minimum, but I do not understand what the question means by at least one critical point is not a given point, and the curve is continuous and differentiable throughout. I understate that piecewise means that the graph can break, but at the point it breaks, the x values are the same, and there must be a hole at that point.

If you could please expand/explain more on the question, I can try solving it. I really need help on this urgently as it is due tomorrow evening. Thanks.
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Edited by stapel -- Reason for edit: Replacing scan of text with text.

 

[skeeter]

Hey guys, this is the question I am trying to solve:

The following five points lie on a function: (1, 20), (2, 4), (5, 3), (6, 2), and (10, 1).

Find an equation which passes through these points, and which also has the following:
. . .i) three inflection points
. . .ii) at least one local maximum
. . .iii) at least one local minimum
. . .iv) at least one critical point which differs from the listed points (above)
. . .v) the property that it is continuous and differentiable throughout
. . .vi) a piecewise definition (that is, it is not a single polynomial)

Now I understand that the graph should have 3 inflection points, one local maximum and one local minimum, but I do not understand what the question means by at least one critical point is not a given point this is telling you that for at least one x = c, f'(c) = 0 and c cannot equal 1, 2, 5, 6, or 10. , and the curve is continuous and differentiable throughout. I understate that piecewise means that the graph can break, but at the point it breaks, the x values are the same, and there must be a hole at that point. no ... you were told the function is continuous and differentiable throughout ... no holes, breaks, or any kind of discontinuities ... the two pieces of the piece-wise function have to "fit" at the same point and with the same slope at that point.

If you could please expand/explain more on the question, I can try solving it. I really need help on this urgently as it is due tomorrow evening. Thanks.
_________________________
Edited by stapel -- Reason for edit: Replacing scan of text with text.