Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/13)  Q#5
Question
A curve has equation . Find the values of x at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.
Solution
Coordinates of stationary point on the curve can be found from the derivative of equation of the curve by equating it with ZERO. This results in value of xcoordinate of the stationary point on the curve.
We are given that;
Therefore first we need gradient of the curve.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Hence;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Hence, to find the x coordinate(s) of stationary points of the curve;
Now we have two options.










Two possible values of imply that there are two stationary points on the curve one at each value of .
Once we have the xcoordinate of the stationary point of a curve, we can determine its nature, whether minimum or maximum, by finding 2^{nd }derivative of the curve.
Second derivative is the derivative of the derivative. If we have derivative of the curve as , then expression for the second derivative of the curve is;
We have found above that;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
We substitute of the stationary point in the expression of 2^{nd} derivative of the curve and evaluate it;
If or then stationary point (or its value) is minimum.
If or then stationary point (or its value) is maximum.
For ; 
For 










Since Minimum 
Since Maximum 
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