Increasing And Decreasing Functions

To determine if the function is increasing or decreasing, we need to find the first derivative and analyze its sign over the intervals in question.

The first derivative will help us determine the intervals where the function is increasing (if ) or decreasing (if ).

Let us begin by differentiating the function:

Given,

Applying product rule to differentiate , where :

First, differentiate using the chain rule:

Now substitute back to find :

On simplifying,

Now, we analyze the sign of in the interval . Since is always positive, the sign of is determined by .

Consider the quadratic .

We can factor or use the discriminant to analyze this quadratic. The roots are found using the quadratic formula:

Where , , and .

So, roots are and .

Quadratic function is an inverted parabola (opens downwards), implying that it is positive between the roots and .

Thus, in the interval , making the function increasing.

This indicates that, within the domain , is increasing.

Hence, the correct answer is that is increasing on .