Question

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

Answer 1

See explanation.

Explanation:

To find if a function is increasing or decreasing at a given point we have to calculate its derivative at the point.

The function is:

`f(x)=-7x^3+x^2-2x-1`

so its derivative is:

`f^'(x)=-21x^2+2x-2`

The value of derivative at `x_0=-2` is:

`f^'(2)=-21*(-2)^2+2*(-2)-2=-84-4-2=-90`

The value of the derivative is less than zero, so the function is decreasing.

Answer 2

Either drawing the graph or taking the derivate `f'(x)=-21x^2+2x-2` in `x=-2` we find that the slope is negative (`f'(-2)=-90`). We can, therefore, conclude that `f(x)` is decreasing at `x=-2`

Explanation:

Let's start by drawing the graph in Geogebra (an excellent, free geometry program):

Looking at it , it is quite clear that `f(x)` is decreasing at `x=-2`

We can also find this without drawing the graph by taking the derivate, which gives the slope of the tangent in each value of x:
`f'(x)=-7*3x^2+2x-2=-21x^2+2x-2`
Insert the value `x=-2`:
`f'(-2)=-21(-2)^2+2(-2)-2=-84-4-2=-90`

As `f'(-2)=-90<0`, we can conclude that `f(-2)` is decreasing in -2.