Question

The function  is defined by  for .

     i.       Find  and  and hence verify that the function  has a minimum value at .

The points  and  lie on the curve , as shown in the diagram.

     i.       Find the distance AB.

   ii.       Find, showing all necessary working, the area of the shaded region.

Solution

i.
 

We are given;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

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

If the given function   has a minimum (stationary) value at  then  at  .

We have found above that for given function ;

At ;

Therefore, function  has a stationary value at .

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 found above that for given function ;

At ;

Since , function  has a minimum value at .

ii.
 

Expression to find distance between two given points  and is:

Therefore for points  and ;

iii.
 

It is evident from the diagram that;

To find the area of region under the curve , we need to integrate the curve from point  to  along x-axis.

We are given equation of the curve as follows;

However, we do not have equation of line but using points  and  we can write  equation of line AB.

Two-Point form of the equation of the line is;

Therefore;

Now we can find area of shaded region.

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is: