# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#3

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**Question**

**
i. **Express in the form , where a, b and c are constants.

** ii. **The function, where , is defined for . Find and state, with a reason, whether is an increasing function, a decreasing function or neither.

**Solution**

i.

We have the expression;

We use method of “completing square” to obtain the desired form. We take out factor ‘3’ from the terms which involve ;

Next we complete the square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

ii.

We have;

Rule for differentiation of is:

Rule for differentiation of is:

Rule for differentiation of is:

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.

As demonstrated in (i), we know that we can write the expression;

Therefore;

It is evident that always;

Hence;

Therefore, function is an increasing function.

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