# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#5

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Question

A curve has equation . Find the values of x at which the curve has a stationary  point and determine the nature of each stationary point, justifying your answers.

Solution

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 are given that;

Therefore first we need gradient of the curve.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

Hence;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Hence, to find the x coordinate(s) of stationary points of the curve;

Now we have two options.

Two possible values of  imply that there are two stationary points on the curve one at each value  of .

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 have found above that;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

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.

 For ; For Since  Minimum Since  Maximum