Question

How do you find the extrema for `f(x)=x^4-18x^2+7`?

Answer 1

This function has 3 extrema:

• A maximum at 0 `f(0)=7`
• 2 minima at -3 and 3 `f(-3)=f(3)=-74`

Explanation:

To calculete the extrema of a function you have to find points, where `f'(x)=0` first.
In this case you get:
`4x^3-36x=0`
`4x(x^2-9)=0`
`x=0 vv x=-3 xx x=3`

Now you have to check how `f'(x)` looks like in the surrounding of the points calculated above.
To check the behaviour you can either draw a graph or calculate `f''(x)`

1. If `f'` changes sign from positive to negative or `f''<0` then it is a maximum
2. If `f'` changes sign from negative to positive or `f''>0`- it is a minimum
3. If `f'` does not change sign then there is no extremum at this point.