# Is #f(x)=(x-1)(x-3)(2x-1)# increasing or decreasing at #x=2#?

##### 1 Answer

It's decreasing.

#### Explanation:

So, to determine whether a function is increasing or decreasing at a point algebraically, the way to go is with a derivative. So, we want to find

I'll save you the work and tell you the the foiled version of

Now, this becomes a very simple power rule derivative. It will end up as:

Before we go any further, we need to establish what the derivative will tell us. Now, by definition, the derivative is the **instantaneous rate of change**, or the **slope of the tangent line** at a point. How does this help us? Well, if our derivative is **positive** at a point, then what would that imply? Well, it would imply that the instantaneous rate of change at that point is positive, or that **the function is increasing** there. We can use the same logic to say that if the derivative is **negative** at a point, the function is **decreasing** there.

Now, all that is left to do is actually take the derivative at

Let's do that:

So, as **decreasing** at

Here's a graph of the function as well, that shows us that we are correct:

graph{(x-1)(x-3)(2x-1) [-10, 10, -5, 5]}

Hope that helped :)