Question

Is `f(x)=-2x^3+2x^2-x+2` increasing or decreasing at `x=0`?

Answer 1

Hey there! Simply put, `f(x)` is decreasing at x = 0. I'll explain why below!

Explanation:

You're looking for a characteristic of this function at x = 0, specifically if it's increasing at a point. With this, you need to complete what is known as the First Derivative Test.

As with any function, to determine if the function is increasing or decreasing at a point, you need the first derivative.

Differentiating we get:

`f'(x) = -6x^2 +4x - 1`

Now you want to substitute the x value you have into the first derivative.

`f'(0) = -6(0)^2 +4(0) - 1`

Simplifying we get:

`f'(0) = -1`

How do we interpret this? What is the meaning of the first derivative?

The first derivative is the slope/rate of change of a function. Thus, since the slope is negative, this means the function is decreasing! Conversely, if the first derivative were to turn out positive, the function would be increasing!

Hopefully this helps! If you have any questions, feel free to ask and I'll do my best to answer them! :)