Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2019  OctNov  (P19709/13)  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;
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