Question

The function f is such that  for , where n is an integer. It is given that  f is an increasing function. Find the least possible value of n.

Solution

We are given function;

We are also given that it is an increasing function.

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:

Since we are given that function is increasing;

Therefore;

To find the set of values of x for which , we solve the following equation to find  critical values of ;

Now we have two options;

Hence the critical points on the curve for the given condition are 3 & -1.

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.

Therefore conditions for  are;

Since we are given that function is increasing, it is only possible if we consider the parabola when  .

Hence, least possible value of  is 3.