Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/13)  Q#4
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Question
The function f is such that for , where n is an integer. It is given that f is an increasing function. Find the least possible value of n.
Solution
We are given function;
We are also given that it is an increasing 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.
Let’s find of the given function.
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Since we are given that function is increasing;
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 options;






Hence the critical points on the curve for the given condition are 3 & 1.
Standard form of quadratic equation is;
The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the graph.
We recognize that given curve , is a parabola opening upwards.
Therefore conditions for are;
Since we are given that function is increasing, it is only possible if we consider the parabola when .
Hence, least possible value of is 3.
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