Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#6

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  Question

A curve is such that . The point (0, 1) lies on the curve.

     i. Find the equation of the curve.

   ii. The curve has one stationary point. Find the x-coordinate of this point and determine whether it  is a maximum or a minimum point.

Solution

     i.
 

We are required to find equation of the curve which has a point (0,1) and;

We can find equation of the curve from its derivative through integration;

Therefore;

Rule for integration of  is:

Rule for integration of , or ;

If a point  lies on the curve , we can find out value of . We substitute values of  and  in the equation obtained from integration of the derivative of the curve i.e. .

Therefore, we substitute coordinates of point (0,1) in the above equation.

Therefore, equation of the curve can be written as;

   ii.
 

We are required to find the x-coordinate of the only stationary point of the curve.

A stationary point on the curve is the point where gradient of the curve is equal to zero; 

We are given that;

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 can find the x-coordinate of the stationary point by equating its gradient with ZERO.

Taking logarithm of both sides;

Power Rule;

Therefore;

Since ;

One possible values of implies that there is only one stationary point on the curve at this value of .

Next we are required to determine the nature of this 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.

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 are already given;

Therefore;

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

Once we have the coordinates 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.

We have already found that point (0,1) is a stationary point on the curve.

We substitute x-coordinate in second derivative expression of the curve obtained above.

Since , the stationary point is minimum.

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