Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/23) | Q#3

Question

The equation of a curve is . Find the exact x-coordinate 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 x-coordinate 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 x-coordinate 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 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 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 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 following are the x-coordinates of two stationary points on the curve.

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

For

For

Maximum

Minimum

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