Question

Is `f(x)=-3x^3-5x^2-x-1` increasing or decreasing at `x=-2`?

Answer 1

Decreasing.

Explanation:

Find the sign of the first derivative at `x=-2:`

• If `f'(-2)<0`, then `f(x)` is decreasing at `x=-2`.
• If `f'(-2)>0`, then `f(x)` is increasing at `x=-2`.

To find the derivative of the function, use the power rule.

`f(x)=-3x^3-5x^2-x-1`

`f'(x)=-9x^2-10x-1`

The sign of the derivative at `x=-2` is

`f'(-2)=-9(-2)^2-10(-2)-1=-36+20-1=ul(-17`

Since this is `<0`, the function is decreasing at `x=-2`. We can check a graph of `f(x):`

graph{-3x^3-5x^2-x-1 [-4, 2, -12, 15]}