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

1 Answer
Jun 16, 2017

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:

enter image source here

The graph may also help you:

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