Recover a cubic from its extrema
The cubic curve has a maximum point and a minimum point . The line segment is the diameter of a circle with equation . Given that the -coordinate of is 1, find the equation of .
[2011/TJC/JCT/JC1/4]
Solution
Modelling with a cubic
The cubic curve has a maximum point and a minimum point . The line segment is a diameter of the circle . Given that the -coordinate of is 1, find the equation of .
Hint
The centre is the midpoint of . Use it to find .
Decide which point is the maximum.
Write : two conditions that the points lie on , two that there.
That puts the minimum above the maximum, which is impossible on the stretch between them.
Two turning points give two conditions as well as two points on the curve: four equations in total.
Answer · Q
.
The other end of the diameter
The centre is the midpoint of the diameter, so .
Heights from the circle
Both and lie on the circle.
Which sign?
is a minimum and a maximum, and is to the right of . Moving from a minimum to the next maximum the curve rises, so must be higher than . That rules out and gives and .
Four equations for
vanishes at and , and passes through both points.
Solve
Subtracting the first two gives , so . Then , , and the last equation gives .
Decimal form
To 3 s.f. This matches the printed answer.
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