# 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

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**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.

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