Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | May-Jun | (P2-9709/02) | Q#6

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  Question

It is given that the curve has one stationary point.

     i.       Find the x-coordinates of this point.

   ii.       Determine whether this point is a maximum or a minimum point.

Solution

     i.
 

We are required to find the coordinates of stationary point of the curve;

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

Since given point is stationary point, therefore, gradient of the curve at this point must ZERO. 

We can find expression for gradient of the curve at stationary point and equate it with ZERO to find  the x-coordinate of this point.

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

Therefore;

If  and  are functions of , and if , then;

If , then;

Let  and ;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

Now we need expression for gradient of the curve at stationary point S.

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Since point S is a stationary point, the gradient of the curve at this point must be equal to ZERO.

Hence, x-coordinate of stationary point S on the curve is .

Corresponding values of y coordinate can be found by substituting values of x in equation of the line  or equation of the curve.

We are given equation of the curve as;

We substitute x=1 in equation of the curve;

Hence coordinates of the stationary point of the curve are .

 

   ii.
 

We are required to find whether this point is a maximum or a minimum point.

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;

From (i) we have;

We can find the second derivative.

First we find .

If  and  are functions of , and if , then;

If , then;

Let  and ;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore;

We have found in (i) that at the stationary point .

If  or then stationary point (or its value) is minimum.

If  or then stationary point (or its value) is maximum.

Therefore, stationary point is a minimum.

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