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.