Question

Is `f(x)=(x^3+3x^2-x-9)/(x+1)` increasing or decreasing at `x=-2`?

Answer 1

Increasing.

Explanation:

To determine if a function is increasing or decreasing at a point, examine the sign of the first derivative at that point.

• 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 `f'(x)`, use the quotient rule.

This gives us:

`f'(x)=((x+1)d/dx(x^3+3x^2-x-9)-(x^3+3x^2-x-9)d/dx(x+1))/(x+1)^2`

Find each derivative through the power rule.

`f'(x)=((x+1)(3x^2+6x-1)-(x^3+3x^2-x-9))/(x+1)^2`

Simplification of the numerator yields:

`f'(x)=(2x^3+6x^2+6x+8)/(x+1)^2`

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

`f'(-2)=(2(-8)+6(4)+6(-2)+8)/(-2+1)^2=4`

Since `f'(2)=4>0`, the function is increasing at `x=-2`. We can check the graph of `f(x)`:

graph{(x^3+3x^2-x-9)/(x+1) [-7.87, 6.18, -1.176, 5.85]}