Question

The diagram shows the curve  defined for x>0. The curve has a minimum point at A and  crosses the x-axis at B and C. It is given that  and that the curve passes through the  point .

i.       Find the x-coordinate of A.

ii.       Find .

iii.       Find the x-coordinates of B and C.

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

Solution

     i.
 

We are given that point A on the curve is a minimum point and hence must be a stationary point on  the curve.

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

We are given that;

Therefore, at point A;

Therefore, x-coordinate of point A is 1.

ii.
 

We are required to find equation of the curve;

We are given that;

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

Therefore;

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 curve passes through a point . 

Substituting these values for x and y in above found equation;

Hence, equation of the curve is;

  iii.
 

We are given that the curve crosses the x-axis at B and C.

Therefore, points B and C are x-intercepts of the curve.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore, utilising equation of the curve as found in (ii);

Let , then , hence;

Now we have two options.

Since ;

It is evident from the diagram that both points B and C are on the positive side of x-axis, therefore;

iv.
 

The area between a curve and the ‘x’ axis is the same regardless whether it is ‘area under the  curve’ or it is ‘area above the curve’.

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.

Equation of the curve was found in (ii) as;

It is evident from the diagram that shaded region extends from  to . Therefore;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is: