# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#2

Hits: 117

Question

It is given that , for . Show that  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.

We are given that;

First we find the derivative of the given function.

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

Let;

Therefore;

Rule for differentiation is of  is:

Rule for differentiation is of  is:

We are given that , for , therefore,  will be never negative and so will be  and .

Hence, will be always positive and  will be always negative.

We can now see that

Therefore, it is  is a decreasing function.