Question

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

Answer 1

The function is decreasing.

Explanation:

The derivative of a function can tell us if the function is increasing or decreasing.

If `f'(2)>0`, then the function is increasing when `x=2`.
If `f'(2)<0`, then the function is decreasing when `x=2`.

To find `f'(x)`, use the power rule.

`f(x)=-x^3-2x^2-3x-1`
`f'(x)=-3x^2-4x-3`

Find `f'(2)`:

`f'(2)=-3(2^2)-4(2)-3=-23`

Since `-23<0`, the function is decreasing when `x=2`.

We can check a graph:

graph{-x^3-2x^2-3x-1 [-16.02, 16.02, -8.01, 8.01]}