Question

Is `f(x)=(-2x^2-4x-2)/(2x^2+1)` increasing or decreasing at `x=-3`?

Answer 1

Increasing.

Explanation:

We compute the slope of the function by finding the first derivative. To do this we shall use the quotient rule:

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

`f'(x) = (-8x^3-4x-8x^2-4 + 8x^3 + 16x^2 + 8x)/((2x^2+1)^2)`

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

`f'(-3) = (8(-3)^2 + 4(-3)-4)/(2(-3)^2+1)^2 = (72-16)/(19)^2 > 0`

`implies` function is increasing, as `f'(x) > 0`