Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2019  FebMar  (P19709/12)  Q#4
Question
A curve has equation .
i. Find and .
ii. Find the xcoordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point.
Solution
i.
We are required to find and .
Therefore, we find the derivative of equation of the curve.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
We are given equation of the curve as;
Therefore;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Next, we are required to find the second derivative .
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 in (i) that;
Therefore;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
ii.
We are required to find the coordinates of the stationary point of the curve.
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 have found in (i) that;
Therefore;
Now we have two options.






Two values of indicate two stationary points on the curve.
Next, we are required to find the nature of each stationary point.
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.
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.
Above, we have found xcoordinates of stationary points on the curve as 0 & 1.
Therefore, substituting and in expression of second derivative we can find nature of both stationary points of the curve.
We have found in (i);
For 
For 














Maximum 
Minimum 
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