# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#8

Hits: 37

Question

A function  is defined for  and is such that .

i.      Find the set of values of  for which f is decreasing.

ii.      It is now given that . Find .

Solution

i.

We are given derivative of the function as;

We are also given that it is a decreasing function.

To test whether a function is increasing or decreasing at a particular point , we  take derivative of a function at that point.

If  , the function  is increasing.

If , the function  is decreasing.

If , the test is inconclusive.

Since we are given that function is decreasing;

Therefore;

To find the set of values of x for which , we solve the following equation to find  critical values of ;

Now we have two critical values;

 Given Found

ii.

We are required to find  from i.e., to find the function from its derivative.

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

We are given;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

We need to find the constant .

We are given that;

Therefore;

Hence;