Question

A curve  has a stationary point at . It is given that, where k is a constant.

i.       Show that  and hence find the x-coordinate of the other stationary point, Q.

   ii.       Find  and determine the nature of each of the stationary points P and Q.

  iii.       Find .

Solution

i.
 

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

Hence gradient of the given curve at given stationary point .

Since;

Therefore;

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.

Now we can write the derivative of the equation of given 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 .

We are already given one stationary point at . So the x coordinate of other stationary point is .

ii.
 

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 given that;

For second derivative;

Rule for differentiation of  is:

Rule for differentiation of  is:

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 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.

Since we have two stationary points namely P & Q with x-coordinates  and  , we check nature of both stationary  points one by one.

For  ;

Since
 it is minimum point.

For  ;

Since  it is maximum point.

iii.
 

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

For the given case;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

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. .

We are given that stationary point at .

Therefore;

Hence;