Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#4
Question
A curve has equation 
.
i. Find 
and 
.
ii. Find the x-coordinates 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 x-coordinate of the stationary point 
on the curve.
We have found in (i) that;
Therefore;
Now we have two options.
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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 x-coordinate of the stationary point of a curve, we can determine its nature, whether minimum or maximum, by finding 2nd derivative of the curve.
We substitute 
of the stationary point
in the expression of 2nd 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 x-coordinates 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);
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Maximum |
Minimum |
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