Question

Is `f(x)=(x^3+5x^2-7x+2)/(x+1)` increasing or decreasing at `x=0`?

Answer 1

Decreasing.

Explanation:

A function is increasing at `x` when its derivative evaluated at `x` is positive. That is, when `f'(x) > 0`. Similarly, a function is decreasing at `x` when `f'(x) < 0`.

In order to test whether our function is increasing or decreasing at `x = 0`, we need to find it's derivative. Note that we use the quotient rule.

`f'(x) = ((x+1)(3x^2 + 10x - 7) - (x^3 + 5x^2 - 7x + 1)) / (x+1)^2`
`= (3x^3 + 10x^2 - 7x + 3x^2 + 10x - 7 - x^3 - 5x^2 + 7x - 1)/(x+1)^2`
`= (2x^3 + 8x^2 + 10x - 8) / (x+1)^2`

Evaluating this at zero gives:

`f'(0) = -8`

Since `f'(0) < 0`, `f(x)` is decreasing at `x = 0`.