# 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

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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; 