Question

How do you ise interval notation indicate where f(x) is concave up and concave down for `f(x)=x^(4)-6x^(3)`?

Answer 1

`f(x)` is concave up from `(-oo,0)uu(3,oo)`
`f(x)` is concave down from `(0,3)`

Explanation:

What you want to do is find the second derivative using the power rule:

`d/dxx^n=nx^(n-1)`

The first derivative is:

`d/dx=4x^3-18x^2`

The second derivative is:

`f''(x)=12x^2-36x`

What you want to do now is factor it and set it equal to zero:

`12x(x-3)=0`

`12x=0` `&` `x-3=0`

`x=0,3`

Now you make a test interval from:
`(-oo,0)uu(0,3)uu(3,oo)`

You test values from the left and right into the second derivative but not the exact values of `x`.
If you get a negative number then it means that at that interval the function is concave down and if it's positive its concave up.

If done so correctly you should get that:

`f(x)` is concave up from `(-oo,0)uu(3,oo)` and that
`f(x)` is concave down from `(0,3)`

You should also note that the points `f(0)` and `f(3)` are inflection points.

Attached below is a picture that may help you:

The graph may also help you:

graph{x^4-6x^3 [-147.3, 145, -125.6, 20.5]}