Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2013  OctNov  (P29709/23)  Q#3
Question
The equation of a curve is . Find the exact xcoordinate of each of the stationary points of the curve and determine the nature of each stationary point.
Solution
First we are required to find the exact xcoordinate of each of the stationary points of the curve.
A stationary point on the curve is the point where gradient of the curve is equal to zero;
Hence, gradient of the curve at stationary point must be ZERO.
We can find expression for gradient of the curve and equate it with ZERO to find the xcoordinate of the point.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Therefore;
Rule for differentiation of is:
Rule for differentiation of natural exponential function , or ;
Rule for differentiation of is:
Since at stationary point the gradient of the curve must be equal to ZERO.
Let ;
Now we have two options.




Since 



Taking logarithm of both sides; 



Since for any 



Since ; 



Two possible values of implies that there are two stationary points on the curve at these values of .
Next we are required to determine the nature of these stationary points.
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 already found;
Therefore;
Rule for differentiation of is:
Rule for differentiation of natural exponential function , or ;
Rule for differentiation of is:
Once we have the coordinates 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.
We have already found that following are the xcoordinates of two stationary points on the curve.


We substitute xcoordinates in second derivative expression of the curve obtained above.
For 
For 












Maximum 



Minimum 
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