# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#4

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Question

The function f is such that  for , where k is a constant. Find the  largest value of k for which f is a decreasing function.

Solution

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.

Therefore, we need to find the derivative of the given equation of the function.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve   with respect to  is:

We are given the function;

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Since we are looking for function as decreasing one;

Since function is defined for ,;

Hence, the largest possible value of  is.