# 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 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 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 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 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.