# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#2

Question

The function f is defined by  for .

Determine, showing all necessary working, whether f is an increasing function, a decreasing  function or neither.

Solution

We are given function;

We are required to find whether is an increasing function, decreasing function or neither.

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.

Let’s find of the given function.

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

We need values of  for for which the original function is defined.

We need to find the solution of the quadratic equation .

We solve the following equation to find critical values of ;

Now we have two options;

Hence the critical points on the curve are -2 & .

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph. If (‘a’ is negative) then parabola opens  downwards and its vertex is the maximum point on the graph.

We recognize that given curve , is a parabola opening upwards.

Since derivative of the  for  ;

for

for

Hence, function is neither increasing nor decreasing.